Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 82.0%

Time: 26.6s

Alternatives: 20

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: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}, 1\right) \cdot \frac{t\_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M\_m \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))) (t_1 (sqrt (- d))))
   (if (<= l -5e-175)
     (*
      (sqrt (/ d l))
      (*
       (/ t_1 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -3.8e-308)
       (*
        (sqrt (/ d h))
        (*
         (fma h (* (/ -0.125 l) (pow (* M_m (/ D d)) 2.0)) 1.0)
         (/ t_1 (sqrt (- l)))))
       (if (<= l 1.7e-120)
         (*
          (/ d t_0)
          (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))
         (*
          d
          (/
           (fma (* -0.125 (pow (/ 1.0 (/ d (* M_m D))) 2.0)) (/ h l) 1.0)
           t_0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -5e-175) {
		tmp = sqrt((d / l)) * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -3.8e-308) {
		tmp = sqrt((d / h)) * (fma(h, ((-0.125 / l) * pow((M_m * (D / d)), 2.0)), 1.0) * (t_1 / sqrt(-l)));
	} else if (l <= 1.7e-120) {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	} else {
		tmp = d * (fma((-0.125 * pow((1.0 / (d / (M_m * D))), 2.0)), (h / l), 1.0) / t_0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -5e-175)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -3.8e-308)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(fma(h, Float64(Float64(-0.125 / l) * (Float64(M_m * Float64(D / d)) ^ 2.0)), 1.0) * Float64(t_1 / sqrt(Float64(-l)))));
	elseif (l <= 1.7e-120)
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(1.0 / Float64(d / Float64(M_m * D))) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -5e-175], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3.8e-308], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e-120], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(1.0 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5e-175

   1. Initial program 67.5%

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

   3. Simplified 67.6%

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

      1. frac-2neg 68.8%

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

      2. sqrt-div 85.3%

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

   6. Applied egg-rr 83.1%

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

   if -5e-175 < l < -3.79999999999999975e-308

   1. Initial program 57.7%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in M around 0 46.2%

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

      1. associate-*r/ 46.2%

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

      2. associate-*r* 46.5%

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

      3. associate-*r* 46.5%

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

      4. associate-*l/ 46.5%

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

      5. associate-*r/ 46.5%

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

      6. *-commutative 46.5%

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

      7. +-commutative 46.5%

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

      8. fma-undefine 46.5%

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

   7. Simplified 71.8%

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

      1. frac-2neg 71.8%

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

      2. sqrt-div 90.6%

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

   9. Applied egg-rr 90.6%

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

   if -3.79999999999999975e-308 < l < 1.70000000000000005e-120

   1. Initial program 60.6%

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

   3. Simplified 53.2%

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

      1. *-commutative 53.2%

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

      2. sqrt-div 55.5%

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

      3. sqrt-div 67.5%

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

      4. frac-times 67.5%

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

      5. add-sqr-sqrt 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. associate-*r/ 87.6%

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

      2. associate-*l/ 87.6%

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

      3. div-inv 87.6%

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

      4. metadata-eval 87.6%

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

      5. *-commutative 87.6%

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

      6. unpow-prod-down 87.6%

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

      7. metadata-eval 87.6%

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

      8. clear-num 85.1%

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

      9. un-div-inv 87.5%

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

   8. Applied egg-rr 87.5%

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

   if 1.70000000000000005e-120 < l

   1. Initial program 57.6%

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

   3. Simplified 57.5%

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

      1. pow1 57.5%

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

      2. associate-*r* 57.5%

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

      3. sqrt-div 69.6%

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

      4. sqrt-div 74.3%

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

      5. frac-times 74.4%

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

      6. add-sqr-sqrt 74.6%

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

      7. unpow-prod-down 74.6%

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

      8. metadata-eval 74.6%

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

      9. clear-num 74.6%

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

      10. un-div-inv 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. unpow1 74.6%

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

      2. associate-*l/ 83.3%

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

      3. associate-/l* 83.3%

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

      4. fma-define 83.3%

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

      5. associate-*r* 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*l* 83.3%

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

      9. metadata-eval 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-/r/ 83.4%

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

      12. associate-*l/ 84.4%

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

      13. associate-/l* 83.3%

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

   8. Simplified 83.3%

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

      1. associate-*r/ 84.4%

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

      2. clear-num 84.4%

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M\_m \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= l -2e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (*
       (sqrt (/ d l))
       (fma h (* (/ -0.125 l) (pow (* M_m (/ D d)) 2.0)) 1.0)))
     (if (<= l 3.7e-121)
       (*
        (/ d t_0)
        (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))
       (*
        d
        (/
         (fma (* -0.125 (pow (/ 1.0 (/ d (* M_m D))) 2.0)) (/ h l) 1.0)
         t_0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * fma(h, ((-0.125 / l) * pow((M_m * (D / d)), 2.0)), 1.0));
	} else if (l <= 3.7e-121) {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	} else {
		tmp = d * (fma((-0.125 * pow((1.0 / (d / (M_m * D))), 2.0)), (h / l), 1.0) / t_0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * fma(h, Float64(Float64(-0.125 / l) * (Float64(M_m * Float64(D / d)) ^ 2.0)), 1.0)));
	elseif (l <= 3.7e-121)
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(1.0 / Float64(d / Float64(M_m * D))) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e-121], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(1.0 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   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 66.0%

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

   5. Taylor expanded in M around 0 41.5%

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

      1. associate-*r/ 41.5%

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

      2. associate-*r* 44.9%

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

      3. associate-*r* 44.9%

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

      4. associate-*l/ 45.7%

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

      5. associate-*r/ 45.7%

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

      6. *-commutative 45.7%

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

      7. +-commutative 45.7%

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

      8. fma-undefine 45.7%

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

   7. Simplified 68.9%

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

      1. frac-2neg 68.9%

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

      2. sqrt-div 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -1.999999999999994e-310 < l < 3.7000000000000002e-121

   1. Initial program 62.1%

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

   3. Simplified 54.5%

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

      1. *-commutative 54.5%

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

      2. sqrt-div 56.9%

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

      3. sqrt-div 69.2%

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

      4. frac-times 69.3%

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

      5. add-sqr-sqrt 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. associate-*r/ 89.9%

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

      2. associate-*l/ 89.9%

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

      3. div-inv 89.9%

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

      4. metadata-eval 89.9%

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

      5. *-commutative 89.9%

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

      6. unpow-prod-down 89.9%

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

      7. metadata-eval 89.9%

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

      8. clear-num 87.3%

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

      9. un-div-inv 89.8%

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

   8. Applied egg-rr 89.8%

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

   if 3.7000000000000002e-121 < l

   1. Initial program 57.6%

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

   3. Simplified 57.5%

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

      1. pow1 57.5%

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

      2. associate-*r* 57.5%

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

      3. sqrt-div 69.6%

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

      4. sqrt-div 74.3%

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

      5. frac-times 74.4%

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

      6. add-sqr-sqrt 74.6%

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

      7. unpow-prod-down 74.6%

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

      8. metadata-eval 74.6%

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

      9. clear-num 74.6%

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

      10. un-div-inv 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. unpow1 74.6%

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

      2. associate-*l/ 83.3%

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

      3. associate-/l* 83.3%

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

      4. fma-define 83.3%

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

      5. associate-*r* 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*l* 83.3%

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

      9. metadata-eval 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-/r/ 83.4%

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

      12. associate-*l/ 84.4%

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

      13. associate-/l* 83.3%

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

   8. Simplified 83.3%

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

      1. associate-*r/ 84.4%

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

      2. clear-num 84.4%

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M\_m}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M\_m}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-116}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M\_m \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -1e+197)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -8.2e-203)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (* 0.5 (* 0.25 (pow (* D (* (/ M_m d) (sqrt (/ h l)))) 2.0)))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (/ (pow M_m 2.0) (- d)))))
         (if (<= d 3.9e-116)
           (*
            d
            (/
             (fma (* -0.125 (pow (/ 1.0 (/ d (* M_m D))) 2.0)) (/ h l) 1.0)
             t_0))
           (*
            (/ d t_0)
            (-
             1.0
             (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -1e+197) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -8.2e-203) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (0.25 * pow((D * ((M_m / d) * sqrt((h / l)))), 2.0))));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) / -d)));
	} else if (d <= 3.9e-116) {
		tmp = d * (fma((-0.125 * pow((1.0 / (d / (M_m * D))), 2.0)), (h / l), 1.0) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -1e+197)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -8.2e-203)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(0.25 * (Float64(D * Float64(Float64(M_m / d) * sqrt(Float64(h / l)))) ^ 2.0)))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) / Float64(-d)))));
	elseif (d <= 3.9e-116)
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(1.0 / Float64(d / Float64(M_m * D))) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e+197], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.2e-203], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(0.25 * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.9e-116], N[(d * N[(N[(N[(-0.125 * N[Power[N[(1.0 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -9.9999999999999995e196

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -9.9999999999999995e196 < d < -8.19999999999999962e-203

   1. Initial program 74.6%

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

   3. Simplified 75.4%

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

      1. add-sqr-sqrt 75.3%

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

      2. pow2 75.3%

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

      3. sqrt-prod 75.4%

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

      4. sqrt-pow1 79.0%

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

      5. metadata-eval 79.0%

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

      6. associate-*l/ 79.0%

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

      7. div-inv 79.0%

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

      8. metadata-eval 79.0%

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

      9. *-commutative 79.0%

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

      10. pow1 79.0%

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

      11. clear-num 78.9%

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

      12. un-div-inv 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. metadata-eval 79.0%

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

      2. pow-prod-up 79.0%

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

      3. associate-*l* 79.0%

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

      4. pow1 79.0%

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

      5. div-inv 78.9%

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

      6. clear-num 78.9%

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

      7. associate-*l* 78.9%

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

      8. pow1 78.9%

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

      9. div-inv 78.9%

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

      10. clear-num 79.0%

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

   8. Applied egg-rr 79.0%

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

      1. swap-sqr 79.0%

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

      2. metadata-eval 79.0%

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

      3. unpow2 79.0%

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

      4. associate-*r/ 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-/l* 77.6%

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

      7. associate-*l* 77.6%

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

   10. Simplified 77.6%

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

   if -8.19999999999999962e-203 < d < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Simplified 37.3%

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

      1. pow1/2 37.3%

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

      2. pow-to-exp 36.5%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. associate-/l* 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      6. rem-square-sqrt 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      7. mul-1-neg 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-{M}^{2}}}{d}\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{-{M}^{2}}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d < 3.9000000000000001e-116

   1. Initial program 42.4%

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

   3. Simplified 40.1%

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

      1. pow1 40.1%

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

      2. associate-*r* 40.1%

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

      3. sqrt-div 49.5%

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

      4. sqrt-div 51.7%

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

      5. frac-times 51.8%

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

      6. add-sqr-sqrt 51.8%

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

      7. unpow-prod-down 51.8%

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

      8. metadata-eval 51.8%

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

      9. clear-num 51.8%

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

      10. un-div-inv 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow1 54.1%

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

      2. associate-*l/ 73.1%

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

      3. associate-/l* 73.1%

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

      4. fma-define 73.1%

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

      5. associate-*r* 73.1%

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

      6. fma-define 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*l* 73.1%

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

      9. metadata-eval 73.1%

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

      10. *-commutative 73.1%

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

      11. associate-/r/ 73.1%

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

      12. associate-*l/ 75.4%

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

      13. associate-/l* 70.9%

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

   8. Simplified 70.9%

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

      1. associate-*r/ 75.4%

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

      2. clear-num 75.4%

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

   10. Applied egg-rr 75.4%

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

   if 3.9000000000000001e-116 < d

   1. Initial program 66.8%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. sqrt-div 69.6%

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

      3. sqrt-div 83.0%

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

      4. frac-times 83.0%

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

      5. add-sqr-sqrt 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-*r/ 91.1%

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

      2. associate-*l/ 91.1%

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

      3. div-inv 91.1%

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

      4. metadata-eval 91.1%

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

      5. *-commutative 91.1%

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

      6. unpow-prod-down 91.1%

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

      7. metadata-eval 91.1%

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

      8. clear-num 90.1%

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

      9. un-div-inv 90.1%

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

   8. Applied egg-rr 90.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-116}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+197}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M\_m}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-113}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M\_m \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -1.9e+197)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -8.5e-203)
       (*
        (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0)))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (/ (pow M_m 2.0) (- d)))))
         (if (<= d 4.5e-113)
           (*
            d
            (/
             (fma (* -0.125 (pow (/ 1.0 (/ d (* M_m D))) 2.0)) (/ h l) 1.0)
             t_0))
           (*
            (/ d t_0)
            (-
             1.0
             (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -1.9e+197) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -8.5e-203) {
		tmp = (sqrt((d / l)) * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) / -d)));
	} else if (d <= 4.5e-113) {
		tmp = d * (fma((-0.125 * pow((1.0 / (d / (M_m * D))), 2.0)), (h / l), 1.0) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -1.9e+197)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -8.5e-203)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) / Float64(-d)))));
	elseif (d <= 4.5e-113)
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(1.0 / Float64(d / Float64(M_m * D))) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.9e+197], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.5e-203], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-113], N[(d * N[(N[(N[(-0.125 * N[Power[N[(1.0 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.9000000000000001e197

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -1.9000000000000001e197 < d < -8.50000000000000031e-203

   1. Initial program 74.6%

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

   3. Simplified 75.4%

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

      1. clear-num 75.3%

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

      2. sqrt-div 77.8%

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

      3. metadata-eval 77.8%

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

   6. Applied egg-rr 77.8%

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

   if -8.50000000000000031e-203 < d < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Simplified 37.3%

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

      1. pow1/2 37.3%

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

      2. pow-to-exp 36.5%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. associate-/l* 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      6. rem-square-sqrt 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      7. mul-1-neg 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-{M}^{2}}}{d}\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{-{M}^{2}}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d < 4.5000000000000001e-113

   1. Initial program 42.4%

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

   3. Simplified 40.1%

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

      1. pow1 40.1%

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

      2. associate-*r* 40.1%

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

      3. sqrt-div 49.5%

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

      4. sqrt-div 51.7%

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

      5. frac-times 51.8%

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

      6. add-sqr-sqrt 51.8%

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

      7. unpow-prod-down 51.8%

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

      8. metadata-eval 51.8%

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

      9. clear-num 51.8%

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

      10. un-div-inv 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow1 54.1%

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

      2. associate-*l/ 73.1%

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

      3. associate-/l* 73.1%

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

      4. fma-define 73.1%

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

      5. associate-*r* 73.1%

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

      6. fma-define 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*l* 73.1%

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

      9. metadata-eval 73.1%

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

      10. *-commutative 73.1%

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

      11. associate-/r/ 73.1%

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

      12. associate-*l/ 75.4%

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

      13. associate-/l* 70.9%

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

   8. Simplified 70.9%

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

      1. associate-*r/ 75.4%

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

      2. clear-num 75.4%

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

   10. Applied egg-rr 75.4%

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

   if 4.5000000000000001e-113 < d

   1. Initial program 66.8%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. sqrt-div 69.6%

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

      3. sqrt-div 83.0%

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

      4. frac-times 83.0%

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

      5. add-sqr-sqrt 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-*r/ 91.1%

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

      2. associate-*l/ 91.1%

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

      3. div-inv 91.1%

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

      4. metadata-eval 91.1%

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

      5. *-commutative 91.1%

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

      6. unpow-prod-down 91.1%

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

      7. metadata-eval 91.1%

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

      8. clear-num 90.1%

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

      9. un-div-inv 90.1%

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

   8. Applied egg-rr 90.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+197}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-113}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+196}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M\_m}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-130}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -3.5e+196)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -8.2e-203)
       (*
        (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0)))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (/ (pow M_m 2.0) (- d)))))
         (if (<= d 1.25e-130)
           (* d (/ (fma (* -0.125 (pow (* M_m (/ D d)) 2.0)) (/ h l) 1.0) t_0))
           (*
            (/ d t_0)
            (-
             1.0
             (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -3.5e+196) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -8.2e-203) {
		tmp = (sqrt((d / l)) * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) / -d)));
	} else if (d <= 1.25e-130) {
		tmp = d * (fma((-0.125 * pow((M_m * (D / d)), 2.0)), (h / l), 1.0) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -3.5e+196)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -8.2e-203)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) / Float64(-d)))));
	elseif (d <= 1.25e-130)
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m * Float64(D / d)) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.5e+196], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.2e-203], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.25e-130], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.4999999999999998e196

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -3.4999999999999998e196 < d < -8.19999999999999962e-203

   1. Initial program 74.6%

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

   3. Simplified 75.4%

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

      1. clear-num 75.3%

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

      2. sqrt-div 77.8%

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

      3. metadata-eval 77.8%

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

   6. Applied egg-rr 77.8%

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

   if -8.19999999999999962e-203 < d < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Simplified 37.3%

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

      1. pow1/2 37.3%

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

      2. pow-to-exp 36.5%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. associate-/l* 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      6. rem-square-sqrt 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      7. mul-1-neg 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-{M}^{2}}}{d}\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{-{M}^{2}}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d < 1.2499999999999999e-130

   1. Initial program 40.5%

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

   3. Simplified 38.1%

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

      1. pow1 38.1%

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

      2. associate-*r* 38.1%

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

      3. sqrt-div 48.2%

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

      4. sqrt-div 50.6%

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

      5. frac-times 50.7%

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

      6. add-sqr-sqrt 50.7%

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

      7. unpow-prod-down 50.7%

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

      8. metadata-eval 50.7%

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

      9. clear-num 50.7%

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

      10. un-div-inv 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. unpow1 53.1%

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

      2. associate-*l/ 73.6%

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

      3. associate-/l* 73.6%

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

      4. fma-define 73.6%

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

      5. associate-*r* 73.6%

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

      6. fma-define 73.6%

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

      7. *-commutative 73.6%

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

      8. associate-*l* 73.6%

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

      9. metadata-eval 73.6%

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

      10. *-commutative 73.6%

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

      11. associate-/r/ 73.6%

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

      12. associate-*l/ 73.6%

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

      13. associate-/l* 71.2%

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

   8. Simplified 71.2%

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

   if 1.2499999999999999e-130 < d

   1. Initial program 66.9%

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

   3. Simplified 64.7%

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

      1. *-commutative 64.7%

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

      2. sqrt-div 69.5%

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

      3. sqrt-div 82.4%

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

      4. frac-times 82.5%

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

      5. add-sqr-sqrt 82.7%

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

   6. Applied egg-rr 82.7%

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

      1. associate-*r/ 90.4%

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

      2. associate-*l/ 90.4%

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

      3. div-inv 90.4%

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

      4. metadata-eval 90.4%

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

      5. *-commutative 90.4%

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

      6. unpow-prod-down 90.4%

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

      7. metadata-eval 90.4%

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

      8. clear-num 89.3%

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

      9. un-div-inv 89.3%

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

   8. Applied egg-rr 89.3%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{+196}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-130}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+198}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M\_m \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -9e+198)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -5e-310)
       (*
        (sqrt (/ d l))
        (*
         (/ (sqrt (- d)) (sqrt (- h)))
         (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
       (if (<= d 1.35e-118)
         (*
          d
          (/
           (fma (* -0.125 (pow (/ 1.0 (/ d (* M_m D))) 2.0)) (/ h l) 1.0)
           t_0))
         (*
          (/ d t_0)
          (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -9e+198) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 1.35e-118) {
		tmp = d * (fma((-0.125 * pow((1.0 / (d / (M_m * D))), 2.0)), (h / l), 1.0) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -9e+198)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= 1.35e-118)
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(1.0 / Float64(d / Float64(M_m * D))) ^ 2.0)), Float64(h / l), 1.0) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9e+198], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -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[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-118], N[(d * N[(N[(N[(-0.125 * N[Power[N[(1.0 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.00000000000000003e198

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -9.00000000000000003e198 < d < -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 65.8%

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

      1. frac-2neg 69.3%

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

      2. sqrt-div 85.9%

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

   6. Applied egg-rr 80.7%

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

   if -4.999999999999985e-310 < d < 1.34999999999999997e-118

   1. Initial program 42.4%

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

   3. Simplified 40.1%

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

      1. pow1 40.1%

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

      2. associate-*r* 40.1%

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

      3. sqrt-div 49.5%

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

      4. sqrt-div 51.7%

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

      5. frac-times 51.8%

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

      6. add-sqr-sqrt 51.8%

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

      7. unpow-prod-down 51.8%

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

      8. metadata-eval 51.8%

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

      9. clear-num 51.8%

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

      10. un-div-inv 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow1 54.1%

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

      2. associate-*l/ 73.1%

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

      3. associate-/l* 73.1%

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

      4. fma-define 73.1%

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

      5. associate-*r* 73.1%

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

      6. fma-define 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*l* 73.1%

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

      9. metadata-eval 73.1%

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

      10. *-commutative 73.1%

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

      11. associate-/r/ 73.1%

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

      12. associate-*l/ 75.4%

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

      13. associate-/l* 70.9%

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

   8. Simplified 70.9%

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

      1. associate-*r/ 75.4%

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

      2. clear-num 75.4%

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

   10. Applied egg-rr 75.4%

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

   if 1.34999999999999997e-118 < d

   1. Initial program 66.8%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. sqrt-div 69.6%

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

      3. sqrt-div 83.0%

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

      4. frac-times 83.0%

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

      5. add-sqr-sqrt 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-*r/ 91.1%

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

      2. associate-*l/ 91.1%

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

      3. div-inv 91.1%

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

      4. metadata-eval 91.1%

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

      5. *-commutative 91.1%

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

      6. unpow-prod-down 91.1%

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

      7. metadata-eval 91.1%

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

      8. clear-num 90.1%

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

      9. un-div-inv 90.1%

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

   8. Applied egg-rr 90.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+198}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{1}{\frac{d}{M \cdot D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M\_m}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -7.4e+196)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -8.5e-203)
       (*
        (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0)))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (/ (pow M_m 2.0) (- d)))))
         (if (<= d 8.6e-197)
           (* d (/ (* h (* (/ -0.125 l) (pow (* M_m (/ D d)) 2.0))) t_0))
           (*
            (/ d t_0)
            (-
             1.0
             (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -7.4e+196) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -8.5e-203) {
		tmp = (sqrt((d / l)) * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) / -d)));
	} else if (d <= 8.6e-197) {
		tmp = d * ((h * ((-0.125 / l) * pow((M_m * (D / d)), 2.0))) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(l) * sqrt(h)
    if (d <= (-7.4d+196)) then
        tmp = -d / sqrt((l * h))
    else if (d <= (-8.5d-203)) then
        tmp = (sqrt((d / l)) * (1.0d0 / sqrt((h / d)))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))))
    else if (d <= (-5d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * ((d_1 ** 2.0d0) * ((m_m ** 2.0d0) / -d)))
    else if (d <= 8.6d-197) then
        tmp = d * ((h * (((-0.125d0) / l) * ((m_m * (d_1 / d)) ** 2.0d0))) / t_0)
    else
        tmp = (d / t_0) * (1.0d0 - (0.5d0 * ((h * (0.25d0 * ((m_m / (d / d_1)) ** 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt(l) * Math.sqrt(h);
	double tmp;
	if (d <= -7.4e+196) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= -8.5e-203) {
		tmp = (Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -5e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (Math.pow(D, 2.0) * (Math.pow(M_m, 2.0) / -d)));
	} else if (d <= 8.6e-197) {
		tmp = d * ((h * ((-0.125 / l) * Math.pow((M_m * (D / d)), 2.0))) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * Math.pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(l) * math.sqrt(h)
	tmp = 0
	if d <= -7.4e+196:
		tmp = -d / math.sqrt((l * h))
	elif d <= -8.5e-203:
		tmp = (math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))))
	elif d <= -5e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (math.pow(D, 2.0) * (math.pow(M_m, 2.0) / -d)))
	elif d <= 8.6e-197:
		tmp = d * ((h * ((-0.125 / l) * math.pow((M_m * (D / d)), 2.0))) / t_0)
	else:
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * math.pow((M_m / (d / D)), 2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -7.4e+196)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -8.5e-203)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) / Float64(-d)))));
	elseif (d <= 8.6e-197)
		tmp = Float64(d * Float64(Float64(h * Float64(Float64(-0.125 / l) * (Float64(M_m * Float64(D / d)) ^ 2.0))) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt(l) * sqrt(h);
	tmp = 0.0;
	if (d <= -7.4e+196)
		tmp = -d / sqrt((l * h));
	elseif (d <= -8.5e-203)
		tmp = (sqrt((d / l)) * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))));
	elseif (d <= -5e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * ((D ^ 2.0) * ((M_m ^ 2.0) / -d)));
	elseif (d <= 8.6e-197)
		tmp = d * ((h * ((-0.125 / l) * ((M_m * (D / d)) ^ 2.0))) / t_0);
	else
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * ((M_m / (d / D)) ^ 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.4e+196], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.5e-203], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.6e-197], N[(d * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -7.3999999999999998e196

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -7.3999999999999998e196 < d < -8.50000000000000031e-203

   1. Initial program 74.6%

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

   3. Simplified 75.4%

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

      1. clear-num 75.3%

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

      2. sqrt-div 77.8%

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

      3. metadata-eval 77.8%

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

   6. Applied egg-rr 77.8%

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

   if -8.50000000000000031e-203 < d < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Simplified 37.3%

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

      1. pow1/2 37.3%

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

      2. pow-to-exp 36.5%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. associate-/l* 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      6. rem-square-sqrt 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      7. mul-1-neg 73.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{-{M}^{2}}}{d}\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{-{M}^{2}}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d < 8.6000000000000001e-197

   1. Initial program 26.3%

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

   3. Simplified 22.1%

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

      1. pow1 22.1%

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

      2. associate-*r* 22.1%

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

      3. sqrt-div 35.3%

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

      4. sqrt-div 35.6%

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

      5. frac-times 35.6%

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

      6. add-sqr-sqrt 35.6%

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

      7. unpow-prod-down 35.6%

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

      8. metadata-eval 35.6%

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

      9. clear-num 35.6%

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

      10. un-div-inv 39.8%

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

   6. Applied egg-rr 39.8%

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

      1. unpow1 39.8%

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

      2. associate-*l/ 70.9%

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

      3. associate-/l* 70.8%

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

      4. fma-define 70.8%

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

      5. associate-*r* 70.8%

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

      6. fma-define 70.8%

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

      7. *-commutative 70.8%

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

      8. associate-*l* 70.8%

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

      9. metadata-eval 70.8%

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

      10. *-commutative 70.8%

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

      11. associate-/r/ 70.7%

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

      12. associate-*l/ 70.8%

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

      13. associate-/l* 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in M around inf 48.7%

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

       1. associate-*r/ 48.7%

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

       2. associate-*r* 48.7%

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

       3. associate-*r* 48.7%

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

       4. associate-*l/ 48.7%

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

       5. associate-*r/ 48.7%

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

       6. *-commutative 48.7%

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

       7. *-commutative 48.7%

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

       8. associate-*l/ 48.7%

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

       9. times-frac 48.7%

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

       10. *-commutative 48.7%

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

       11. associate-/l* 49.0%

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

       12. unpow2 49.0%

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

       13. unpow2 49.0%

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

       14. unpow2 49.0%

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

       15. times-frac 62.0%

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

       16. swap-sqr 66.7%

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

       17. unpow2 66.7%

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

   11. Simplified 66.7%

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

   if 8.6000000000000001e-197 < d

   1. Initial program 65.8%

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

   3. Simplified 63.9%

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

      1. *-commutative 63.9%

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

      2. sqrt-div 68.8%

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

      3. sqrt-div 80.7%

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

      4. frac-times 80.7%

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

      5. add-sqr-sqrt 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. associate-*r/ 87.4%

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

      2. associate-*l/ 87.4%

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

      3. div-inv 87.4%

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

      4. metadata-eval 87.4%

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

      5. *-commutative 87.4%

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

      6. unpow-prod-down 87.4%

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

      7. metadata-eval 87.4%

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

      8. clear-num 86.5%

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

      9. un-div-inv 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-203}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{-d}\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ t_1 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{+198}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-43}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{t\_0}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot t\_0\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_1} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0)) (t_1 (* (sqrt l) (sqrt h))))
   (if (<= d -5.6e+198)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -2.1e-43)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (* 0.5 (* h (* 0.25 (/ t_0 l))))))
       (if (<= d -5e-310)
         (*
          (* d (sqrt (/ 1.0 (* l h))))
          (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
         (if (<= d 1.2e-197)
           (* d (/ (* h (* (/ -0.125 l) t_0)) t_1))
           (*
            (/ d t_1)
            (-
             1.0
             (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double t_1 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -5.6e+198) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -2.1e-43) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (d <= 1.2e-197) {
		tmp = d * ((h * ((-0.125 / l) * t_0)) / t_1);
	} else {
		tmp = (d / t_1) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * (d_1 / d)) ** 2.0d0
    t_1 = sqrt(l) * sqrt(h)
    if (d <= (-5.6d+198)) then
        tmp = -d / sqrt((l * h))
    else if (d <= (-2.1d-43)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (h * (0.25d0 * (t_0 / l)))))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (d <= 1.2d-197) then
        tmp = d * ((h * (((-0.125d0) / l) * t_0)) / t_1)
    else
        tmp = (d / t_1) * (1.0d0 - (0.5d0 * ((h * (0.25d0 * ((m_m / (d / d_1)) ** 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((M_m * (D / d)), 2.0);
	double t_1 = Math.sqrt(l) * Math.sqrt(h);
	double tmp;
	if (d <= -5.6e+198) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= -2.1e-43) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (d <= 1.2e-197) {
		tmp = d * ((h * ((-0.125 / l) * t_0)) / t_1);
	} else {
		tmp = (d / t_1) * (1.0 - (0.5 * ((h * (0.25 * Math.pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((M_m * (D / d)), 2.0)
	t_1 = math.sqrt(l) * math.sqrt(h)
	tmp = 0
	if d <= -5.6e+198:
		tmp = -d / math.sqrt((l * h))
	elif d <= -2.1e-43:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif d <= 1.2e-197:
		tmp = d * ((h * ((-0.125 / l) * t_0)) / t_1)
	else:
		tmp = (d / t_1) * (1.0 - (0.5 * ((h * (0.25 * math.pow((M_m / (d / D)), 2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	t_1 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -5.6e+198)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -2.1e-43)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64(t_0 / l))))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (d <= 1.2e-197)
		tmp = Float64(d * Float64(Float64(h * Float64(Float64(-0.125 / l) * t_0)) / t_1));
	else
		tmp = Float64(Float64(d / t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D / d)) ^ 2.0;
	t_1 = sqrt(l) * sqrt(h);
	tmp = 0.0;
	if (d <= -5.6e+198)
		tmp = -d / sqrt((l * h));
	elseif (d <= -2.1e-43)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (d <= 1.2e-197)
		tmp = d * ((h * ((-0.125 / l) * t_0)) / t_1);
	else
		tmp = (d / t_1) * (1.0 - (0.5 * ((h * (0.25 * ((M_m / (d / D)) ^ 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.6e+198], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.1e-43], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e-197], N[(d * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.59999999999999999e198

   1. Initial program 54.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in M around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. associate-*r* 43.9%

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

      3. associate-*r* 43.9%

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

      4. associate-*l/ 43.9%

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

      5. associate-*r/ 43.9%

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

      6. *-commutative 43.9%

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

      7. +-commutative 43.9%

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

      8. fma-undefine 43.9%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.2%

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

      5. neg-mul-1 92.2%

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

      6. distribute-rgt-neg-in 92.2%

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

      7. unpow1/2 92.2%

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

      8. rem-exp-log 90.3%

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

      9. exp-neg 90.3%

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

      10. exp-prod 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. exp-neg 90.3%

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

      13. exp-to-pow 92.2%

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

      14. unpow1/2 92.2%

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

      15. associate-*l/ 92.2%

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

      16. *-lft-identity 92.2%

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

      17. distribute-frac-neg2 92.2%

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

   10. Simplified 92.2%

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

   if -5.59999999999999999e198 < d < -2.1000000000000001e-43

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in M around 0 49.9%

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

      1. associate-*r* 51.9%

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

      2. times-frac 52.0%

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

      3. associate-/l* 52.0%

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

      4. *-commutative 52.0%

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

      5. unpow2 52.0%

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

      6. unpow2 52.0%

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

      7. times-frac 64.9%

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

      8. unpow2 64.9%

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

      9. swap-sqr 75.9%

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

      10. associate-/r/ 75.9%

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

      11. associate-/r/ 77.8%

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

      12. unpow2 77.8%

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

      13. associate-*l* 77.8%

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

      14. *-commutative 77.8%

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

      15. associate-*l/ 77.9%

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

      16. associate-/l* 81.4%

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

      17. associate-/l* 81.4%

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

   7. Simplified 81.4%

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

   if -2.1000000000000001e-43 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Simplified 55.0%

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

      1. clear-num 54.9%

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

      2. sqrt-div 54.9%

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

      3. metadata-eval 54.9%

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

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in d around -inf 69.4%

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

   if -4.999999999999985e-310 < d < 1.2e-197

   1. Initial program 26.3%

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

   3. Simplified 22.1%

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

      1. pow1 22.1%

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

      2. associate-*r* 22.1%

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

      3. sqrt-div 35.3%

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

      4. sqrt-div 35.6%

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

      5. frac-times 35.6%

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

      6. add-sqr-sqrt 35.6%

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

      7. unpow-prod-down 35.6%

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

      8. metadata-eval 35.6%

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

      9. clear-num 35.6%

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

      10. un-div-inv 39.8%

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

   6. Applied egg-rr 39.8%

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

      1. unpow1 39.8%

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

      2. associate-*l/ 70.9%

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

      3. associate-/l* 70.8%

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

      4. fma-define 70.8%

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

      5. associate-*r* 70.8%

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

      6. fma-define 70.8%

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

      7. *-commutative 70.8%

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

      8. associate-*l* 70.8%

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

      9. metadata-eval 70.8%

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

      10. *-commutative 70.8%

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

      11. associate-/r/ 70.7%

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

      12. associate-*l/ 70.8%

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

      13. associate-/l* 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in M around inf 48.7%

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

       1. associate-*r/ 48.7%

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

       2. associate-*r* 48.7%

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

       3. associate-*r* 48.7%

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

       4. associate-*l/ 48.7%

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

       5. associate-*r/ 48.7%

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

       6. *-commutative 48.7%

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

       7. *-commutative 48.7%

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

       8. associate-*l/ 48.7%

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

       9. times-frac 48.7%

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

       10. *-commutative 48.7%

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

       11. associate-/l* 49.0%

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

       12. unpow2 49.0%

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

       13. unpow2 49.0%

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

       14. unpow2 49.0%

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

       15. times-frac 62.0%

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

       16. swap-sqr 66.7%

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

       17. unpow2 66.7%

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

   11. Simplified 66.7%

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

   if 1.2e-197 < d

   1. Initial program 65.8%

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

   3. Simplified 63.9%

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

      1. *-commutative 63.9%

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

      2. sqrt-div 68.8%

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

      3. sqrt-div 80.7%

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

      4. frac-times 80.7%

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

      5. add-sqr-sqrt 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. associate-*r/ 87.4%

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

      2. associate-*l/ 87.4%

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

      3. div-inv 87.4%

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

      4. metadata-eval 87.4%

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

      5. *-commutative 87.4%

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

      6. unpow-prod-down 87.4%

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

      7. metadata-eval 87.4%

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

      8. clear-num 86.5%

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

      9. un-div-inv 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{+198}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-43}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-198}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (sqrt l) (sqrt h))))
   (if (<= d -4.5e+199)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
       (if (<= d 7.6e-198)
         (* d (/ (* h (* (/ -0.125 l) (pow (* M_m (/ D d)) 2.0))) t_0))
         (*
          (/ d t_0)
          (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D)) 2.0))) l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -4.5e+199) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (d <= 7.6e-198) {
		tmp = d * ((h * ((-0.125 / l) * pow((M_m * (D / d)), 2.0))) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(l) * sqrt(h)
    if (d <= (-4.5d+199)) then
        tmp = -d / sqrt((l * h))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (d <= 7.6d-198) then
        tmp = d * ((h * (((-0.125d0) / l) * ((m_m * (d_1 / d)) ** 2.0d0))) / t_0)
    else
        tmp = (d / t_0) * (1.0d0 - (0.5d0 * ((h * (0.25d0 * ((m_m / (d / d_1)) ** 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt(l) * Math.sqrt(h);
	double tmp;
	if (d <= -4.5e+199) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (d <= 7.6e-198) {
		tmp = d * ((h * ((-0.125 / l) * Math.pow((M_m * (D / d)), 2.0))) / t_0);
	} else {
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * Math.pow((M_m / (d / D)), 2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(l) * math.sqrt(h)
	tmp = 0
	if d <= -4.5e+199:
		tmp = -d / math.sqrt((l * h))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif d <= 7.6e-198:
		tmp = d * ((h * ((-0.125 / l) * math.pow((M_m * (D / d)), 2.0))) / t_0)
	else:
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * math.pow((M_m / (d / D)), 2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -4.5e+199)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (d <= 7.6e-198)
		tmp = Float64(d * Float64(Float64(h * Float64(Float64(-0.125 / l) * (Float64(M_m * Float64(D / d)) ^ 2.0))) / t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D)) ^ 2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt(l) * sqrt(h);
	tmp = 0.0;
	if (d <= -4.5e+199)
		tmp = -d / sqrt((l * h));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (d <= 7.6e-198)
		tmp = d * ((h * ((-0.125 / l) * ((M_m * (D / d)) ^ 2.0))) / t_0);
	else
		tmp = (d / t_0) * (1.0 - (0.5 * ((h * (0.25 * ((M_m / (d / D)) ^ 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.5e+199], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e-198], N[(d * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.4999999999999997e199

   1. Initial program 56.7%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in M around 0 37.2%

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

      1. associate-*r/ 37.2%

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

      2. associate-*r* 45.4%

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

      3. associate-*r* 45.4%

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

      4. associate-*l/ 45.4%

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

      5. associate-*r/ 45.4%

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

      6. *-commutative 45.4%

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

      7. +-commutative 45.4%

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

      8. fma-undefine 45.4%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 91.9%

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

      5. neg-mul-1 91.9%

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

      6. distribute-rgt-neg-in 91.9%

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

      7. unpow1/2 91.9%

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

      8. rem-exp-log 90.0%

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

      9. exp-neg 90.1%

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

      10. exp-prod 90.1%

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

      11. distribute-lft-neg-out 90.1%

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

      12. exp-neg 90.1%

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

      13. exp-to-pow 91.9%

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

      14. unpow1/2 91.9%

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

      15. associate-*l/ 91.9%

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

      16. *-lft-identity 91.9%

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

      17. distribute-frac-neg2 91.9%

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

   10. Simplified 91.9%

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

   if -4.4999999999999997e199 < d < -4.999999999999985e-310

   1. Initial program 66.7%

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

   3. Simplified 67.3%

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

      1. clear-num 67.2%

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

      2. sqrt-div 67.2%

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

      3. metadata-eval 67.2%

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

   6. Applied egg-rr 67.2%

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

   7. Taylor expanded in d around -inf 70.5%

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

   if -4.999999999999985e-310 < d < 7.6000000000000004e-198

   1. Initial program 26.3%

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

   3. Simplified 22.1%

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

      1. pow1 22.1%

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

      2. associate-*r* 22.1%

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

      3. sqrt-div 35.3%

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

      4. sqrt-div 35.6%

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

      5. frac-times 35.6%

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

      6. add-sqr-sqrt 35.6%

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

      7. unpow-prod-down 35.6%

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

      8. metadata-eval 35.6%

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

      9. clear-num 35.6%

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

      10. un-div-inv 39.8%

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

   6. Applied egg-rr 39.8%

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

      1. unpow1 39.8%

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

      2. associate-*l/ 70.9%

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

      3. associate-/l* 70.8%

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

      4. fma-define 70.8%

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

      5. associate-*r* 70.8%

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

      6. fma-define 70.8%

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

      7. *-commutative 70.8%

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

      8. associate-*l* 70.8%

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

      9. metadata-eval 70.8%

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

      10. *-commutative 70.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 70.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 70.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 66.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around inf 48.7%

      \[\leadsto d \cdot \frac{\color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]
   10. Step-by-step derivation

       1. associate-*r/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       2. associate-*r* 48.7%

          \[\leadsto d \cdot \frac{\frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       3. associate-*r* 48.7%

          \[\leadsto d \cdot \frac{\frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       4. associate-*l/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       5. associate-*r/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h}{\sqrt{h} \cdot \sqrt{\ell}} \]

       6. *-commutative 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       7. *-commutative 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} \cdot -0.125\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       8. associate-*l/ 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -0.125}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       9. times-frac 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       10. *-commutative 48.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       11. associate-/l* 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       12. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       13. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       14. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. times-frac 62.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       16. swap-sqr 66.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       17. unpow2 66.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. Simplified 66.7%

       \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   if 7.6000000000000004e-198 < d

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 68.8%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 80.7%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 80.7%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 80.9%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. associate-*l/ 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. div-inv 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. metadata-eval 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{0.5}\right)}^{2} \cdot h}{\ell}\right) \]

      5. *-commutative 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{2} \cdot h}{\ell}\right) \]

      6. unpow-prod-down 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\left({0.5}^{2} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      7. metadata-eval 87.4%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      8. clear-num 86.5%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      9. un-div-inv 86.5%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   8. Applied egg-rr 86.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-198}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\\ t_1 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-196}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_1} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))))
        (t_1 (* (sqrt l) (sqrt h))))
   (if (<= d -3.6e+199)
     (/ (- d) (sqrt (* l h)))
     (if (<= d -5e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (if (<= d 3.2e-196)
         (* d (/ (* h (* (/ -0.125 l) (pow (* M_m (/ D d)) 2.0))) t_1))
         (* (/ d t_1) (- 1.0 t_0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0));
	double t_1 = sqrt(l) * sqrt(h);
	double tmp;
	if (d <= -3.6e+199) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (d <= 3.2e-196) {
		tmp = d * ((h * ((-0.125 / l) * pow((M_m * (D / d)), 2.0))) / t_1);
	} else {
		tmp = (d / t_1) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))
    t_1 = sqrt(l) * sqrt(h)
    if (d <= (-3.6d+199)) then
        tmp = -d / sqrt((l * h))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else if (d <= 3.2d-196) then
        tmp = d * ((h * (((-0.125d0) / l) * ((m_m * (d_1 / d)) ** 2.0d0))) / t_1)
    else
        tmp = (d / t_1) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0));
	double t_1 = Math.sqrt(l) * Math.sqrt(h);
	double tmp;
	if (d <= -3.6e+199) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (d <= 3.2e-196) {
		tmp = d * ((h * ((-0.125 / l) * Math.pow((M_m * (D / d)), 2.0))) / t_1);
	} else {
		tmp = (d / t_1) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))
	t_1 = math.sqrt(l) * math.sqrt(h)
	tmp = 0
	if d <= -3.6e+199:
		tmp = -d / math.sqrt((l * h))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	elif d <= 3.2e-196:
		tmp = d * ((h * ((-0.125 / l) * math.pow((M_m * (D / d)), 2.0))) / t_1)
	else:
		tmp = (d / t_1) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))
	t_1 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (d <= -3.6e+199)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	elseif (d <= 3.2e-196)
		tmp = Float64(d * Float64(Float64(h * Float64(Float64(-0.125 / l) * (Float64(M_m * Float64(D / d)) ^ 2.0))) / t_1));
	else
		tmp = Float64(Float64(d / t_1) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0));
	t_1 = sqrt(l) * sqrt(h);
	tmp = 0.0;
	if (d <= -3.6e+199)
		tmp = -d / sqrt((l * h));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	elseif (d <= 3.2e-196)
		tmp = d * ((h * ((-0.125 / l) * ((M_m * (D / d)) ^ 2.0))) / t_1);
	else
		tmp = (d / t_1) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+199], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-196], N[(d * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$1), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.60000000000000001e199

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 37.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 70.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 91.9%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 91.9%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 91.9%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 91.9%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 90.0%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 90.1%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 90.1%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 90.1%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 90.1%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 91.9%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 91.9%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 91.9%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 91.9%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 91.9%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 91.9%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -3.60000000000000001e199 < d < -4.999999999999985e-310

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 67.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 70.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < d < 3.2e-196

   1. Initial program 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 22.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 22.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 22.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-div 35.3%

         \[\leadsto {\left(\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. sqrt-div 35.6%

         \[\leadsto {\left(\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. frac-times 35.6%

         \[\leadsto {\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      6. add-sqr-sqrt 35.6%

         \[\leadsto {\left(\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      7. unpow-prod-down 35.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{{0.5}^{2} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      8. metadata-eval 35.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 35.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 39.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 39.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 39.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 70.9%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 70.8%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 70.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 70.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 70.8%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 70.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 70.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 70.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 70.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 70.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 70.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 66.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around inf 48.7%

      \[\leadsto d \cdot \frac{\color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]
   10. Step-by-step derivation

       1. associate-*r/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       2. associate-*r* 48.7%

          \[\leadsto d \cdot \frac{\frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       3. associate-*r* 48.7%

          \[\leadsto d \cdot \frac{\frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       4. associate-*l/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       5. associate-*r/ 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h}{\sqrt{h} \cdot \sqrt{\ell}} \]

       6. *-commutative 48.7%

          \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       7. *-commutative 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} \cdot -0.125\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       8. associate-*l/ 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -0.125}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       9. times-frac 48.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       10. *-commutative 48.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       11. associate-/l* 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       12. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       13. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       14. unpow2 49.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. times-frac 62.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       16. swap-sqr 66.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       17. unpow2 66.7%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. Simplified 66.7%

       \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   if 3.2e-196 < d

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 68.8%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 80.7%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 80.7%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 80.9%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-196}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -7.5e+199)
     (/ (- d) t_0)
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
       (*
        d
        (/
         (fma -0.5 (* h (/ (pow (* D (/ (/ M_m d) 2.0)) 2.0) l)) 1.0)
         t_0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -7.5e+199) {
		tmp = -d / t_0;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = d * (fma(-0.5, (h * (pow((D * ((M_m / d) / 2.0)), 2.0) / l)), 1.0) / t_0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -7.5e+199)
		tmp = Float64(Float64(-d) / t_0);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(d * Float64(fma(-0.5, Float64(h * Float64((Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0) / l)), 1.0) / t_0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -7.5e+199], N[((-d) / t$95$0), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.49999999999999977e199

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 37.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 45.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 70.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 91.9%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 91.9%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 91.9%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 91.9%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 90.0%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 90.1%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 90.1%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 90.1%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 90.1%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 91.9%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 91.9%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 91.9%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 91.9%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 91.9%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 91.9%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -7.49999999999999977e199 < d < -4.999999999999985e-310

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 67.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 70.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < d

   1. Initial program 58.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 60.8%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 72.9%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 72.9%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 73.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 79.2%

         \[\leadsto \color{blue}{\frac{d \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      2. cancel-sign-sub-inv 79.2%

         \[\leadsto \frac{d \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      3. metadata-eval 79.2%

         \[\leadsto \frac{d \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      4. *-commutative 79.2%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      5. associate-*l/ 79.2%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. sqrt-unprod 63.4%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. *-commutative 63.4%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   8. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/l* 63.3%

         \[\leadsto \color{blue}{d \cdot \frac{1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}} \]

      2. +-commutative 63.3%

         \[\leadsto d \cdot \frac{\color{blue}{-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right) + 1}}{\sqrt{h \cdot \ell}} \]

      3. fma-define 63.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}, 1\right)}}{\sqrt{h \cdot \ell}} \]

      4. associate-*l/ 70.0%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, \color{blue}{\frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}}, 1\right)}{\sqrt{h \cdot \ell}} \]

      5. associate-/l* 70.1%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, \color{blue}{h \cdot \frac{{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}}, 1\right)}{\sqrt{h \cdot \ell}} \]

      6. associate-*r/ 72.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{h \cdot \ell}} \]

      7. *-commutative 72.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(\frac{\frac{\color{blue}{D \cdot M}}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{h \cdot \ell}} \]

      8. associate-/l* 71.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(\frac{\color{blue}{D \cdot \frac{M}{d}}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{h \cdot \ell}} \]

      9. associate-/l* 71.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\color{blue}{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}}^{2}}{\ell}, 1\right)}{\sqrt{h \cdot \ell}} \]

   10. Simplified 71.6%

       \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+187}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{t\_0}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {t\_0}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D d))))
   (if (<= h -4e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
     (if (<= h 1.35e+187)
       (/
        (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (/ t_0 2.0) 2.0)))))
        (sqrt (* l h)))
       (* d (/ (* h (* (/ -0.125 l) (pow t_0 2.0))) (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / d);
	double tmp;
	if (h <= -4e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (h <= 1.35e+187) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow((t_0 / 2.0), 2.0))))) / sqrt((l * h));
	} else {
		tmp = d * ((h * ((-0.125 / l) * pow(t_0, 2.0))) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_1 / d)
    if (h <= (-4d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (h <= 1.35d+187) then
        tmp = (d * (1.0d0 + ((-0.5d0) * ((h / l) * ((t_0 / 2.0d0) ** 2.0d0))))) / sqrt((l * h))
    else
        tmp = d * ((h * (((-0.125d0) / l) * (t_0 ** 2.0d0))) / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / d);
	double tmp;
	if (h <= -4e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (h <= 1.35e+187) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow((t_0 / 2.0), 2.0))))) / Math.sqrt((l * h));
	} else {
		tmp = d * ((h * ((-0.125 / l) * Math.pow(t_0, 2.0))) / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = M_m * (D / d)
	tmp = 0
	if h <= -4e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif h <= 1.35e+187:
		tmp = (d * (1.0 + (-0.5 * ((h / l) * math.pow((t_0 / 2.0), 2.0))))) / math.sqrt((l * h))
	else:
		tmp = d * ((h * ((-0.125 / l) * math.pow(t_0, 2.0))) / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d))
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (h <= 1.35e+187)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(t_0 / 2.0) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(Float64(h * Float64(Float64(-0.125 / l) * (t_0 ^ 2.0))) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = M_m * (D / d);
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (h <= 1.35e+187)
		tmp = (d * (1.0 + (-0.5 * ((h / l) * ((t_0 / 2.0) ^ 2.0))))) / sqrt((l * h));
	else
		tmp = d * ((h * ((-0.125 / l) * (t_0 ^ 2.0))) / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.35e+187], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(t$95$0 / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.999999999999988e-310

   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 66.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 70.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -3.999999999999988e-310 < h < 1.35000000000000004e187

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 66.5%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 79.3%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 79.4%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 79.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 83.6%

         \[\leadsto \color{blue}{\frac{d \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      2. cancel-sign-sub-inv 83.6%

         \[\leadsto \frac{d \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      3. metadata-eval 83.6%

         \[\leadsto \frac{d \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      4. *-commutative 83.6%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      5. associate-*l/ 83.6%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. sqrt-unprod 72.7%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. *-commutative 72.7%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{h \cdot \ell}}} \]

   if 1.35000000000000004e187 < h

   1. Initial program 45.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 42.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 42.0%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-div 51.7%

         \[\leadsto {\left(\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. sqrt-div 51.7%

         \[\leadsto {\left(\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. frac-times 51.7%

         \[\leadsto {\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      6. add-sqr-sqrt 51.7%

         \[\leadsto {\left(\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      7. unpow-prod-down 51.7%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{{0.5}^{2} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      8. metadata-eval 51.7%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 48.5%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 48.5%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 48.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 61.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 61.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 61.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 61.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 61.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 61.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 61.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 61.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 61.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 68.1%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 68.1%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 64.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 64.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around inf 33.4%

      \[\leadsto d \cdot \frac{\color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]
   10. Step-by-step derivation

       1. associate-*r/ 33.4%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       2. associate-*r* 33.6%

          \[\leadsto d \cdot \frac{\frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       3. associate-*r* 33.6%

          \[\leadsto d \cdot \frac{\frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       4. associate-*l/ 33.7%

          \[\leadsto d \cdot \frac{\color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       5. associate-*r/ 33.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h}{\sqrt{h} \cdot \sqrt{\ell}} \]

       6. *-commutative 33.7%

          \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       7. *-commutative 33.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} \cdot -0.125\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       8. associate-*l/ 33.7%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -0.125}{{d}^{2} \cdot \ell}}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       9. times-frac 37.1%

          \[\leadsto d \cdot \frac{h \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       10. *-commutative 37.1%

           \[\leadsto d \cdot \frac{h \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       11. associate-/l* 37.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       12. unpow2 37.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       13. unpow2 37.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       14. unpow2 37.0%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. times-frac 43.4%

           \[\leadsto d \cdot \frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       16. swap-sqr 56.1%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

       17. unpow2 56.1%

           \[\leadsto d \cdot \frac{h \cdot \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. Simplified 56.1%

       \[\leadsto d \cdot \frac{\color{blue}{h \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+187}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+166}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
   (if (<= l 6e+166)
     (/
      (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m (/ D d)) 2.0) 2.0)))))
      (sqrt (* l h)))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 6e+166) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow(((M_m * (D / d)) / 2.0), 2.0))))) / sqrt((l * h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 6d+166) then
        tmp = (d * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * (d_1 / d)) / 2.0d0) ** 2.0d0))))) / sqrt((l * h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 6e+166) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * (D / d)) / 2.0), 2.0))))) / Math.sqrt((l * h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif l <= 6e+166:
		tmp = (d * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * (D / d)) / 2.0), 2.0))))) / math.sqrt((l * h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (l <= 6e+166)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (l <= 6e+166)
		tmp = (d * (1.0 + (-0.5 * ((h / l) * (((M_m * (D / d)) / 2.0) ^ 2.0))))) / sqrt((l * h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+166], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   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 66.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 70.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -1.999999999999994e-310 < l < 5.99999999999999997e166

   1. Initial program 63.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 60.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 61.8%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 73.5%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 73.5%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 73.7%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{d \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      2. cancel-sign-sub-inv 78.9%

         \[\leadsto \frac{d \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      3. metadata-eval 78.9%

         \[\leadsto \frac{d \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      4. *-commutative 78.9%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      5. associate-*l/ 78.9%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. sqrt-unprod 70.8%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   8. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{h \cdot \ell}}} \]

   if 5.99999999999999997e166 < l

   1. Initial program 47.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 48.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 48.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 48.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 48.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 62.0%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 62.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 62.1%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+166}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.125 \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -7.2e+60)
   (* (- d) (pow (* l h) -0.5))
   (if (<= d -2e-274)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* (/ h l) (* 0.125 (pow (* D (/ M_m d)) 2.0)))))
     (/
      (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m (/ D d)) 2.0) 2.0)))))
      (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.2e+60) {
		tmp = -d * pow((l * h), -0.5);
	} else if (d <= -2e-274) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow(((M_m * (D / d)) / 2.0), 2.0))))) / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= (-7.2d+60)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (d <= (-2d-274)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h / l) * (0.125d0 * ((d_1 * (m_m / d)) ** 2.0d0))))
    else
        tmp = (d * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * (d_1 / d)) / 2.0d0) ** 2.0d0))))) / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.2e+60) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (d <= -2e-274) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * Math.pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * (D / d)) / 2.0), 2.0))))) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -7.2e+60:
		tmp = -d * math.pow((l * h), -0.5)
	elif d <= -2e-274:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * math.pow((D * (M_m / d)), 2.0))))
	else:
		tmp = (d * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * (D / d)) / 2.0), 2.0))))) / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -7.2e+60)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (d <= -2e-274)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.125 * (Float64(D * Float64(M_m / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0))))) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -7.2e+60)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (d <= -2e-274)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * ((D * (M_m / d)) ^ 2.0))));
	else
		tmp = (d * (1.0 + (-0.5 * ((h / l) * (((M_m * (D / d)) / 2.0) ^ 2.0))))) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -7.2e+60], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-274], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.125 * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.19999999999999935e60

   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 69.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      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 75.1%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 75.1%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow1/2 75.1%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      6. rem-exp-log 71.6%

         \[\leadsto d \cdot \left(-{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5}\right) \]

      7. exp-neg 71.6%

         \[\leadsto d \cdot \left(-{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5}\right) \]

      8. exp-prod 71.6%

         \[\leadsto d \cdot \left(-\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}}\right) \]

      9. distribute-lft-neg-out 71.6%

         \[\leadsto d \cdot \left(-e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}}\right) \]

      10. distribute-rgt-neg-in 71.6%

          \[\leadsto d \cdot \left(-e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}}\right) \]

      11. metadata-eval 71.6%

          \[\leadsto d \cdot \left(-e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}}\right) \]

      12. exp-to-pow 75.1%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 75.1%

      \[\leadsto d \cdot \color{blue}{\left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -7.19999999999999935e60 < d < -1.99999999999999993e-274

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l/ 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{0.5}\right)}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. pow1 71.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. clear-num 71.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. un-div-inv 71.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 71.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{\color{blue}{\left(1 + 1\right)}}\right) \]

      2. pow-prod-up 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)}\right) \]

      3. associate-*l* 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{1} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      4. pow1 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      5. div-inv 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      6. clear-num 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \color{blue}{\frac{D}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      7. associate-*l* 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{1}\right)\right) \]

      8. pow1 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}\right)\right) \]

      9. div-inv 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

      10. clear-num 71.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\left(M \cdot \color{blue}{\frac{D}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

   8. Applied egg-rr 71.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. swap-sqr 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.5 \cdot 0.5\right) \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)}\right) \]

      2. metadata-eval 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0.25} \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

      3. unpow2 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right) \]

      4. associate-*r/ 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\color{blue}{\frac{M \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      5. *-commutative 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      6. associate-/l* 70.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      7. associate-*l* 70.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right)\right) \]

   10. Simplified 70.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. pow1 70.1%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)}^{1}} \]

       2. sqrt-unprod 62.1%

          \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)}^{1} \]

       3. associate-*r* 62.1%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(0.5 \cdot 0.25\right) \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right)\right)}^{1} \]

       4. metadata-eval 62.1%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125} \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)}^{1} \]

       5. associate-*r* 62.2%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot {\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right)\right)}^{1} \]

       6. unpow-prod-down 57.5%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)\right)}^{1} \]

       7. pow2 57.5%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)}\right)\right)\right)}^{1} \]

       8. add-sqr-sqrt 57.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 57.6%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 57.6%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. *-commutative 57.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. associate-*r* 57.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

   14. Simplified 57.6%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

   if -1.99999999999999993e-274 < d

   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 55.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 55.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 59.0%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 70.7%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 70.8%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 71.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \color{blue}{\frac{d \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      2. cancel-sign-sub-inv 76.9%

         \[\leadsto \frac{d \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      3. metadata-eval 76.9%

         \[\leadsto \frac{d \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      4. *-commutative 76.9%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      5. associate-*l/ 76.9%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2}\right)\right)}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. sqrt-unprod 61.5%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. *-commutative 61.5%

         \[\leadsto \frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   8. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 10^{+101}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.125 \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -7.2e+60)
   (* (- d) (pow (* l h) -0.5))
   (if (<= d 1e+101)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* (/ h l) (* 0.125 (pow (* D (/ M_m d)) 2.0)))))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.2e+60) {
		tmp = -d * pow((l * h), -0.5);
	} else if (d <= 1e+101) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= (-7.2d+60)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (d <= 1d+101) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h / l) * (0.125d0 * ((d_1 * (m_m / d)) ** 2.0d0))))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.2e+60) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (d <= 1e+101) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * Math.pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -7.2e+60:
		tmp = -d * math.pow((l * h), -0.5)
	elif d <= 1e+101:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * math.pow((D * (M_m / d)), 2.0))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -7.2e+60)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (d <= 1e+101)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.125 * (Float64(D * Float64(M_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -7.2e+60)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (d <= 1e+101)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.125 * ((D * (M_m / d)) ^ 2.0))));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -7.2e+60], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e+101], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.125 * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.19999999999999935e60

   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 69.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      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 75.1%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 75.1%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow1/2 75.1%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      6. rem-exp-log 71.6%

         \[\leadsto d \cdot \left(-{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5}\right) \]

      7. exp-neg 71.6%

         \[\leadsto d \cdot \left(-{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5}\right) \]

      8. exp-prod 71.6%

         \[\leadsto d \cdot \left(-\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}}\right) \]

      9. distribute-lft-neg-out 71.6%

         \[\leadsto d \cdot \left(-e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}}\right) \]

      10. distribute-rgt-neg-in 71.6%

          \[\leadsto d \cdot \left(-e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}}\right) \]

      11. metadata-eval 71.6%

          \[\leadsto d \cdot \left(-e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}}\right) \]

      12. exp-to-pow 75.1%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 75.1%

      \[\leadsto d \cdot \color{blue}{\left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -7.19999999999999935e60 < d < 9.9999999999999998e100

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 58.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l/ 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{0.5}\right)}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. pow1 61.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. clear-num 61.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. un-div-inv 61.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 61.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{\color{blue}{\left(1 + 1\right)}}\right) \]

      2. pow-prod-up 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)}\right) \]

      3. associate-*l* 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{1} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      4. pow1 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)} \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      5. div-inv 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      6. clear-num 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \color{blue}{\frac{D}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{1}\right)\right) \]

      7. associate-*l* 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot {\color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{1}\right)\right) \]

      8. pow1 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}\right)\right) \]

      9. div-inv 61.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

      10. clear-num 61.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\left(M \cdot \color{blue}{\frac{D}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

   8. Applied egg-rr 61.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(0.5 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. swap-sqr 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.5 \cdot 0.5\right) \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)}\right) \]

      2. metadata-eval 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0.25} \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)\right)\right) \]

      3. unpow2 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right) \]

      4. associate-*r/ 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\color{blue}{\frac{M \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      5. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      6. associate-/l* 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      7. associate-*l* 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. pow1 60.5%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)}^{1}} \]

       2. sqrt-unprod 51.9%

          \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)}^{1} \]

       3. associate-*r* 51.9%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(0.5 \cdot 0.25\right) \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right)\right)}^{1} \]

       4. metadata-eval 51.9%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125} \cdot {\left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)}^{1} \]

       5. associate-*r* 51.8%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot {\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right)\right)}^{1} \]

       6. unpow-prod-down 48.7%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)\right)}^{1} \]

       7. pow2 48.7%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)}\right)\right)\right)}^{1} \]

       8. add-sqr-sqrt 48.7%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 48.7%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 48.7%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. *-commutative 48.7%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. associate-*r* 48.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

   14. Simplified 48.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

   if 9.9999999999999998e100 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 60.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 61.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 61.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 61.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 73.3%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 73.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 65.6%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 73.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 73.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 10^{+101}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left({\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -5.2e+39)
   (/ (- d) (sqrt (* l h)))
   (if (<= d 1.02e-131)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (* (pow (* D (/ M_m d)) 2.0) (* -0.125 (/ h l))))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -5.2e+39) {
		tmp = -d / sqrt((l * h));
	} else if (d <= 1.02e-131) {
		tmp = sqrt(((d / l) * (d / h))) * (pow((D * (M_m / d)), 2.0) * (-0.125 * (h / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= (-5.2d+39)) then
        tmp = -d / sqrt((l * h))
    else if (d <= 1.02d-131) then
        tmp = sqrt(((d / l) * (d / h))) * (((d_1 * (m_m / d)) ** 2.0d0) * ((-0.125d0) * (h / l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -5.2e+39) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= 1.02e-131) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (Math.pow((D * (M_m / d)), 2.0) * (-0.125 * (h / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -5.2e+39:
		tmp = -d / math.sqrt((l * h))
	elif d <= 1.02e-131:
		tmp = math.sqrt(((d / l) * (d / h))) * (math.pow((D * (M_m / d)), 2.0) * (-0.125 * (h / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -5.2e+39)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= 1.02e-131)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(-0.125 * Float64(h / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -5.2e+39)
		tmp = -d / sqrt((l * h));
	elseif (d <= 1.02e-131)
		tmp = sqrt(((d / l) * (d / h))) * (((D * (M_m / d)) ^ 2.0) * (-0.125 * (h / l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -5.2e+39], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-131], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.2e39

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Taylor expanded in M around 0 37.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 37.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 74.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 74.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 74.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 74.4%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 74.4%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 70.9%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 71.0%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 71.0%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 71.0%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 71.0%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 74.4%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 74.4%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 74.4%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 74.4%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 74.4%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 74.4%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.2e39 < d < 1.02000000000000001e-131

   1. Initial program 53.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 53.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 53.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l/ 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{0.5}\right)}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{1} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. pow1 57.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. clear-num 57.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. un-div-inv 58.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 58.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in M around inf 28.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   8. Step-by-step derivation

      1. associate-/l* 30.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. times-frac 31.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-*r* 29.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      4. associate-/l* 29.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}} \cdot \frac{h}{\ell}\right)\right) \]

      5. *-commutative 29.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right)\right) \]

      6. associate-/l* 29.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right) \]

      7. unpow2 29.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. unpow2 29.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. unpow2 29.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac 34.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. swap-sqr 38.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right)\right) \]

      12. unpow2 38.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right) \]

      13. *-commutative 38.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}\right) \]

      14. associate-*r/ 39.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right) \]

      15. *-commutative 39.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right) \]

      16. associate-/l* 38.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right) \]

   9. Simplified 38.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. pow1 38.6%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)}^{1}} \]

       2. sqrt-unprod 33.6%

          \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)}^{1} \]

       3. associate-*r* 33.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)}^{1} \]

   11. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 33.6%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} \]

       2. *-commutative 33.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \]

       3. *-commutative 33.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right)\right)} \]

   13. Simplified 33.6%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right)\right)} \]

   if 1.02000000000000001e-131 < d

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 51.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 52.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 52.5%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 52.5%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 61.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 62.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 62.1%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(\ell \cdot h + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -5.5e-52)
   (/ (- d) (sqrt (* l h)))
   (if (<= d -5e-310)
     (* d (exp (* (log1p (+ (* l h) -1.0)) (- 0.5))))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -5.5e-52) {
		tmp = -d / sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * exp((log1p(((l * h) + -1.0)) * -0.5));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -5.5e-52) {
		tmp = -d / Math.sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((Math.log1p(((l * h) + -1.0)) * -0.5));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -5.5e-52:
		tmp = -d / math.sqrt((l * h))
	elif d <= -5e-310:
		tmp = d * math.exp((math.log1p(((l * h) + -1.0)) * -0.5))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -5.5e-52)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(log1p(Float64(Float64(l * h) + -1.0)) * Float64(-0.5))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -5.5e-52], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(N[Log[1 + N[(N[(l * h), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * (-0.5)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.5e-52

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 44.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 44.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 48.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 48.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 49.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 49.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 49.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 49.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 49.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 76.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 62.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 62.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 62.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 62.2%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 59.1%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 59.1%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 59.1%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 59.1%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 59.1%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 62.2%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 62.2%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 62.3%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 62.3%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 62.3%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 62.3%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.5e-52 < d < -4.999999999999985e-310

   1. Initial program 53.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 12.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1/2 12.5%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

      2. pow-to-exp 12.5%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 10.1%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   7. Applied egg-rr 10.1%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 39.0%

         \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

      2. expm1-undefine 39.0%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

      3. add-exp-log 39.0%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   9. Applied egg-rr 39.0%

      \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if -4.999999999999985e-310 < d

   1. Initial program 58.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 41.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 41.6%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 41.6%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 48.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 49.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 46.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 49.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 49.0%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(\ell \cdot h + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 9.6e-200) (/ (- d) (sqrt (* l h))) (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 9.6e-200) {
		tmp = -d / sqrt((l * h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= 9.6d-200) then
        tmp = -d / sqrt((l * h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 9.6e-200) {
		tmp = -d / Math.sqrt((l * h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 9.6e-200:
		tmp = -d / math.sqrt((l * h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 9.6e-200)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 9.6e-200)
		tmp = -d / sqrt((l * h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 9.6e-200], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.60000000000000006e-200

   1. Initial program 59.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 37.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 40.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 40.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 44.7%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 44.7%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 44.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 44.7%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 42.6%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 42.6%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 42.6%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 42.6%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 42.6%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 44.7%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 44.7%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 44.7%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 44.7%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 44.7%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 44.7%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 9.60000000000000006e-200 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 47.8%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 47.8%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 47.8%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 56.5%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 56.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 53.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 56.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 56.6%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= l 4.1e-225) (/ (- d) t_0) (/ d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= 4.1e-225) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= 4.1d-225) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= 4.1e-225) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= 4.1e-225:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= 4.1e-225)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= 4.1e-225)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 4.1e-225], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.10000000000000022e-225

   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 66.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 40.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 40.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 43.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 43.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 44.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 44.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 44.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 44.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 44.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 69.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 47.1%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 47.1%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

      6. distribute-rgt-neg-in 47.1%

         \[\leadsto \color{blue}{-\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. unpow1/2 47.1%

         \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      8. rem-exp-log 44.8%

         \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      9. exp-neg 44.8%

         \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      10. exp-prod 44.8%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      11. distribute-lft-neg-out 44.8%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      12. exp-neg 44.9%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      13. exp-to-pow 47.0%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      14. unpow1/2 47.0%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      15. associate-*l/ 47.1%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      16. *-lft-identity 47.1%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      17. distribute-frac-neg2 47.1%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 4.10000000000000022e-225 < l

   1. Initial program 56.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 33.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 33.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. associate-*r* 33.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. associate-*r* 33.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      4. associate-*l/ 34.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

      5. associate-*r/ 34.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

      6. *-commutative 34.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. +-commutative 34.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

      8. fma-undefine 34.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

   7. Simplified 56.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

   8. Taylor expanded in d around inf 45.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. unpow1/2 45.2%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

      3. rem-exp-log 43.1%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      4. exp-neg 43.1%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      5. exp-prod 43.1%

         \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      6. distribute-lft-neg-out 43.1%

         \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      7. exp-neg 43.1%

         \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      8. exp-to-pow 45.3%

         \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      9. unpow1/2 45.3%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      10. unpow-1 45.3%

          \[\leadsto \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

      11. unpow-1 45.3%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

      12. associate-*l/ 45.4%

          \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      13. *-lft-identity 45.4%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 45.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 37.5%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ 37.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)\right) \]

   2. associate-*r* 39.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

   3. associate-*r* 39.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

   4. associate-*l/ 40.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot h}\right)\right) \]

   5. associate-*r/ 40.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \cdot h\right)\right) \]

   6. *-commutative 40.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

   7. +-commutative 40.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + 1\right)}\right) \]

   8. fma-undefine 40.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, 1\right)}\right) \]

7. Simplified 63.5%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)}\right) \]

8. Taylor expanded in d around inf 24.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. *-commutative 24.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. unpow1/2 24.6%

      \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

   3. rem-exp-log 23.7%

      \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

   4. exp-neg 23.7%

      \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

   5. exp-prod 23.5%

      \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

   6. distribute-lft-neg-out 23.5%

      \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

   7. exp-neg 23.5%

      \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

   8. exp-to-pow 24.5%

      \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

   9. unpow1/2 24.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

   10. unpow-1 24.5%

       \[\leadsto \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

   11. unpow-1 24.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

   12. associate-*l/ 24.6%

       \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

   13. *-lft-identity 24.6%

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

10. Simplified 24.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

11. Final simplification 24.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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)))))