Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 82.7%

Time: 23.9s

Alternatives: 19

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: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.75 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{h \cdot \left(-0.5 \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}\right)}{\ell} + 1\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -1.75e+86)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ (/ (* h (* -0.5 (pow (* M_m (/ D_m (* d 2.0))) 2.0))) l) 1.0)))
   (if (<= h -2e-310)
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (+ (* 0.5 (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ h l))) -1.0))
     (*
      d
      (/
       (fma h (* -0.5 (/ (pow (* D_m (* 0.5 (/ M_m d))) 2.0) l)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -1.75e+86) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (((h * (-0.5 * pow((M_m * (D_m / (d * 2.0))), 2.0))) / l) + 1.0));
	} else if (h <= -2e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l))) + -1.0);
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((D_m * (0.5 * (M_m / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -1.75e+86)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(Float64(Float64(h * Float64(-0.5 * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0))) / l) + 1.0)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))) + -1.0));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -1.75e+86], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(-0.5 * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.75000000000000009e86

   1. Initial program 54.2%

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

   3. Simplified 52.1%

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

      1. frac-2neg 52.1%

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

      2. sqrt-div 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. associate-*l/ 79.5%

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

      2. *-commutative 79.5%

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

      3. associate-*r/ 77.5%

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

      4. associate-*l/ 77.3%

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

      5. *-commutative 77.3%

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

      6. frac-times 77.5%

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

      7. associate-/l* 77.3%

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

      8. *-commutative 77.3%

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

   8. Applied egg-rr 77.3%

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

   if -1.75000000000000009e86 < h < -1.999999999999994e-310

   1. Initial program 77.0%

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

   3. Simplified 75.8%

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

      1. clear-num 75.8%

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

      2. un-div-inv 75.8%

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

      3. frac-times 77.1%

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

      4. associate-/l* 75.8%

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

      5. *-commutative 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-/r/ 74.1%

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

   8. Simplified 74.1%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 85.5%

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

       7. neg-mul-1 85.5%

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

   11. Simplified 85.5%

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

       1. pow1 85.5%

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

       2. associate-*l/ 86.8%

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

       3. associate-/r/ 86.1%

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

   13. Applied egg-rr 86.1%

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

       1. unpow1 86.1%

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

       2. associate-/l* 86.0%

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

       3. associate-*l/ 87.2%

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

   15. Simplified 87.2%

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

   if -1.999999999999994e-310 < h

   1. Initial program 61.7%

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

   3. Simplified 60.8%

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

   6. Applied egg-rr 66.9%

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

   8. Simplified 83.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.75 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell} + 1\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right) + 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.5e-75)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))) 1.0)))
   (if (<= l -4e-310)
     (*
      (* d (pow (* h l) -0.5))
      (+ (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l))) -1.0))
     (*
      d
      (/
       (fma h (* -0.5 (/ (pow (* D_m (* 0.5 (/ M_m d))) 2.0) l)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.5e-75) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0))) + 1.0));
	} else if (l <= -4e-310) {
		tmp = (d * pow((h * l), -0.5)) * ((0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((D_m * (0.5 * (M_m / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.5e-75)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))) + 1.0)));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(d * (Float64(h * l) ^ -0.5)) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.5e-75], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4999999999999999e-75

   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 63.3%

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

      1. frac-2neg 63.3%

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

      2. sqrt-div 76.0%

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

   6. Applied egg-rr 76.0%

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

   if -1.4999999999999999e-75 < l < -3.999999999999988e-310

   1. Initial program 72.1%

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

   3. Simplified 69.9%

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

      1. clear-num 70.0%

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

      2. un-div-inv 70.0%

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

      3. frac-times 72.1%

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

      4. associate-/l* 70.0%

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

      5. *-commutative 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. associate-/r/ 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/r/ 74.8%

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

   8. Simplified 74.8%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 89.6%

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

       7. neg-mul-1 89.6%

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

   11. Simplified 89.6%

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

   12. Taylor expanded in l around 0 89.6%

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

       1. unpow1/2 89.6%

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

       2. rem-exp-log 87.7%

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

       3. exp-neg 87.7%

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

       4. exp-prod 87.7%

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

       5. distribute-lft-neg-out 87.7%

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

       6. distribute-rgt-neg-in 87.7%

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

       7. metadata-eval 87.7%

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

       8. exp-to-pow 89.7%

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

   14. Simplified 89.7%

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

   if -3.999999999999988e-310 < l

   1. Initial program 61.7%

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

   3. Simplified 60.8%

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

   6. Applied egg-rr 66.9%

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

   8. Simplified 83.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right) + 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-205}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -6.6e-205)
   (*
    (* d (sqrt (/ (/ 1.0 l) h)))
    (+ (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l))) -1.0))
   (if (<= d -2e-310)
     (*
      d
      (*
       (sqrt (/ 1.0 (* h l)))
       (- -1.0 (* -0.5 (* (/ h l) (pow (* (* M_m 0.5) (/ D_m d)) 2.0))))))
     (*
      d
      (/
       (fma h (* -0.5 (/ (pow (* D_m (* 0.5 (/ M_m d))) 2.0) l)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -6.6e-205) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (d <= -2e-310) {
		tmp = d * (sqrt((1.0 / (h * l))) * (-1.0 - (-0.5 * ((h / l) * pow(((M_m * 0.5) * (D_m / d)), 2.0)))));
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((D_m * (0.5 * (M_m / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -6.6e-205)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0));
	elseif (d <= -2e-310)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -6.6e-205], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.5999999999999998e-205

   1. Initial program 76.2%

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

   3. Simplified 74.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.4%

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

      3. frac-times 76.3%

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

      4. associate-/l* 74.4%

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

      5. *-commutative 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-/r/ 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 82.3%

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

       7. neg-mul-1 82.3%

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

   11. Simplified 82.3%

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

   if -6.5999999999999998e-205 < d < -1.999999999999994e-310

   1. Initial program 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in d around 0 6.1%

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

      1. *-commutative 6.1%

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

   7. Simplified 6.1%

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

      1. pow1 6.1%

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

      2. pow1/2 6.1%

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

      3. inv-pow 6.1%

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

      4. pow-pow 6.1%

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

      5. metadata-eval 6.1%

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

      6. cancel-sign-sub-inv 6.1%

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

      7. metadata-eval 6.1%

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

      8. *-commutative 6.1%

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

      9. div-inv 6.1%

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

      10. metadata-eval 6.1%

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

   9. Applied egg-rr 6.1%

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

       1. unpow1 6.1%

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

       2. associate-*l* 6.1%

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

       3. *-commutative 6.1%

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

   11. Simplified 6.1%

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

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 66.1%

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

       4. mul-1-neg 66.1%

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

       5. *-commutative 66.1%

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

   14. Simplified 66.1%

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

   if -1.999999999999994e-310 < d

   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 60.8%

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

   6. Applied egg-rr 66.9%

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

   8. Simplified 83.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-205}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;d \leq -6 \cdot 10^{-205}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l)))))
   (if (<= d -6e-205)
     (* (* d (sqrt (/ (/ 1.0 l) h))) (+ t_0 -1.0))
     (if (<= d 1.9e-307)
       (*
        d
        (*
         (sqrt (/ 1.0 (* h l)))
         (- -1.0 (* -0.5 (* (/ h l) (pow (* (* M_m 0.5) (/ D_m d)) 2.0))))))
       (* (- 1.0 t_0) (/ d (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l));
	double tmp;
	if (d <= -6e-205) {
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	} else if (d <= 1.9e-307) {
		tmp = d * (sqrt((1.0 / (h * l))) * (-1.0 - (-0.5 * ((h / l) * pow(((M_m * 0.5) * (D_m / d)), 2.0)))));
	} else {
		tmp = (1.0 - t_0) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m / ((d * 2.0d0) / m_m)) ** 2.0d0) / l))
    if (d <= (-6d-205)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (t_0 + (-1.0d0))
    else if (d <= 1.9d-307) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * (((m_m * 0.5d0) * (d_m / d)) ** 2.0d0)))))
    else
        tmp = (1.0d0 - t_0) * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l));
	double tmp;
	if (d <= -6e-205) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	} else if (d <= 1.9e-307) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow(((M_m * 0.5) * (D_m / d)), 2.0)))));
	} else {
		tmp = (1.0 - t_0) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))
	tmp = 0
	if d <= -6e-205:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (t_0 + -1.0)
	elif d <= 1.9e-307:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (-1.0 - (-0.5 * ((h / l) * math.pow(((M_m * 0.5) * (D_m / d)), 2.0)))))
	else:
		tmp = (1.0 - t_0) * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l)))
	tmp = 0.0
	if (d <= -6e-205)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(t_0 + -1.0));
	elseif (d <= 1.9e-307)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m / ((d * 2.0) / M_m)) ^ 2.0) / l));
	tmp = 0.0;
	if (d <= -6e-205)
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	elseif (d <= 1.9e-307)
		tmp = d * (sqrt((1.0 / (h * l))) * (-1.0 - (-0.5 * ((h / l) * (((M_m * 0.5) * (D_m / d)) ^ 2.0)))));
	else
		tmp = (1.0 - t_0) * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6e-205], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-307], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6e-205

   1. Initial program 76.2%

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

   3. Simplified 74.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.4%

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

      3. frac-times 76.3%

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

      4. associate-/l* 74.4%

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

      5. *-commutative 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-/r/ 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 82.3%

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

       7. neg-mul-1 82.3%

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

   11. Simplified 82.3%

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

   if -6e-205 < d < 1.89999999999999993e-307

   1. Initial program 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in d around 0 5.8%

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

      1. *-commutative 5.8%

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

   7. Simplified 5.8%

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

      1. pow1 5.8%

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

      2. pow1/2 5.8%

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

      3. inv-pow 5.8%

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

      4. pow-pow 5.8%

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

      5. metadata-eval 5.8%

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

      6. cancel-sign-sub-inv 5.8%

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

      7. metadata-eval 5.8%

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

      8. *-commutative 5.8%

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

      9. div-inv 5.8%

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

      10. metadata-eval 5.8%

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

   9. Applied egg-rr 5.8%

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

       1. unpow1 5.8%

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

       2. associate-*l* 10.6%

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

       3. *-commutative 10.6%

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

   11. Simplified 10.6%

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

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 62.9%

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

       4. mul-1-neg 62.9%

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

       5. *-commutative 62.9%

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

   14. Simplified 62.9%

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

   if 1.89999999999999993e-307 < d

   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 61.3%

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

      1. clear-num 61.3%

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

      2. un-div-inv 61.3%

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

      3. frac-times 62.1%

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

      4. associate-/l* 61.3%

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

      5. *-commutative 61.3%

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

   6. Applied egg-rr 61.3%

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

      1. associate-/r/ 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-/r/ 63.1%

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

   8. Simplified 63.1%

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

      1. *-commutative 63.1%

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

      2. sqrt-div 72.6%

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

      3. sqrt-div 80.4%

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

      4. frac-times 80.4%

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

      5. add-sqr-sqrt 80.6%

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

   10. Applied egg-rr 80.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-205}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}\\ t_1 := 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(t\_1 + -1\right)\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-117}:\\ \;\;\;\;\left(1 - t\_1\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(t\_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (/ D_m (/ (* d 2.0) M_m)) 2.0))
        (t_1 (* 0.5 (* h (/ t_0 l)))))
   (if (<= l -4e-310)
     (* (* d (sqrt (/ (/ 1.0 l) h))) (+ t_1 -1.0))
     (if (<= l 7.6e-117)
       (* (- 1.0 t_1) (* d (pow (* h l) -0.5)))
       (* (/ d (* (sqrt l) (sqrt h))) (+ (* t_0 (* -0.5 (/ h l))) 1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((D_m / ((d * 2.0) / M_m)), 2.0);
	double t_1 = 0.5 * (h * (t_0 / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_1 + -1.0);
	} else if (l <= 7.6e-117) {
		tmp = (1.0 - t_1) * (d * pow((h * l), -0.5));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * ((t_0 * (-0.5 * (h / l))) + 1.0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m / ((d * 2.0d0) / m_m)) ** 2.0d0
    t_1 = 0.5d0 * (h * (t_0 / l))
    if (l <= (-4d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (t_1 + (-1.0d0))
    else if (l <= 7.6d-117) then
        tmp = (1.0d0 - t_1) * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * ((t_0 * ((-0.5d0) * (h / l))) + 1.0d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((D_m / ((d * 2.0) / M_m)), 2.0);
	double t_1 = 0.5 * (h * (t_0 / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (t_1 + -1.0);
	} else if (l <= 7.6e-117) {
		tmp = (1.0 - t_1) * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * ((t_0 * (-0.5 * (h / l))) + 1.0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((D_m / ((d * 2.0) / M_m)), 2.0)
	t_1 = 0.5 * (h * (t_0 / l))
	tmp = 0
	if l <= -4e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (t_1 + -1.0)
	elif l <= 7.6e-117:
		tmp = (1.0 - t_1) * (d * math.pow((h * l), -0.5))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * ((t_0 * (-0.5 * (h / l))) + 1.0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0
	t_1 = Float64(0.5 * Float64(h * Float64(t_0 / l)))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(t_1 + -1.0));
	elseif (l <= 7.6e-117)
		tmp = Float64(Float64(1.0 - t_1) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(Float64(t_0 * Float64(-0.5 * Float64(h / l))) + 1.0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / ((d * 2.0) / M_m)) ^ 2.0;
	t_1 = 0.5 * (h * (t_0 / l));
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_1 + -1.0);
	elseif (l <= 7.6e-117)
		tmp = (1.0 - t_1) * (d * ((h * l) ^ -0.5));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * ((t_0 * (-0.5 * (h / l))) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.6e-117], N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.2%

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

   3. Simplified 66.6%

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

      1. clear-num 66.6%

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

      2. un-div-inv 66.6%

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

      3. frac-times 68.2%

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

      4. associate-/l* 66.6%

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

      5. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-/r/ 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 76.3%

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

       7. neg-mul-1 76.3%

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

   11. Simplified 76.3%

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

   if -3.999999999999988e-310 < l < 7.59999999999999945e-117

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. un-div-inv 72.3%

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

      3. frac-times 72.3%

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

      4. associate-/l* 72.3%

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

      5. *-commutative 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. associate-/r/ 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-/r/ 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 0.4%

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

       7. neg-mul-1 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 94.5%

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

       4. neg-mul-1 94.5%

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

       5. unpow1/2 94.5%

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

       6. rem-exp-log 93.2%

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

       7. exp-neg 93.2%

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

       8. exp-prod 93.2%

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

       9. distribute-lft-neg-out 93.2%

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

       10. distribute-rgt-neg-in 93.2%

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

       11. metadata-eval 93.2%

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

       12. exp-to-pow 94.6%

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

   14. Simplified 94.6%

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

   if 7.59999999999999945e-117 < l

   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 56.5%

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

   6. Applied egg-rr 71.1%

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

      1. *-rgt-identity 71.1%

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

      2. distribute-lft-in 74.3%

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

      3. associate-*r* 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-117}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left({\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4e-310)
   (*
    (* d (sqrt (/ (/ 1.0 l) h)))
    (+ (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l))) -1.0))
   (if (<= l 1.12e+236)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (fma h (* -0.5 (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)) 1.0)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (l <= 1.12e+236) {
		tmp = d * (pow((h * l), -0.5) * fma(h, (-0.5 * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l)), 1.0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0));
	elseif (l <= 1.12e+236)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * fma(h, Float64(-0.5 * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)), 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.12e+236], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(h * N[(-0.5 * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.2%

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

   3. Simplified 66.6%

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

      1. clear-num 66.6%

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

      2. un-div-inv 66.6%

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

      3. frac-times 68.2%

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

      4. associate-/l* 66.6%

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

      5. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-/r/ 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 76.3%

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

       7. neg-mul-1 76.3%

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

   11. Simplified 76.3%

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

   if -3.999999999999988e-310 < l < 1.1200000000000001e236

   1. Initial program 66.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in d around 0 72.0%

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

      1. *-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. pow1 72.0%

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

      2. pow1/2 72.0%

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

      3. inv-pow 72.0%

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

      4. pow-pow 72.6%

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

      5. metadata-eval 72.6%

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

      6. cancel-sign-sub-inv 72.6%

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

      7. metadata-eval 72.6%

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

      8. *-commutative 72.6%

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

      9. div-inv 72.6%

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

      10. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

   11. Simplified 81.6%

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

   if 1.1200000000000001e236 < l

   1. Initial program 36.6%

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

   3. Simplified 36.3%

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

   6. Applied egg-rr 45.4%

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

   8. Simplified 45.8%

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

   9. Taylor expanded in d around inf 37.5%

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

   10. Taylor expanded in d around 0 46.9%

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

       1. unpow1/2 46.9%

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

       2. rem-exp-log 43.5%

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

       3. exp-neg 43.5%

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

       4. exp-prod 43.5%

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

       5. distribute-lft-neg-out 43.5%

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

       6. distribute-rgt-neg-in 43.5%

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

       7. metadata-eval 43.5%

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

       8. exp-to-pow 46.8%

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

   12. Simplified 46.8%

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

       1. *-commutative 46.8%

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

       2. unpow-prod-down 65.7%

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

   14. Applied egg-rr 65.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4e-310)
   (*
    (* d (sqrt (/ (/ 1.0 l) h)))
    (+ (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l))) -1.0))
   (if (<= l 2.1e+236)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (+ (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)) 1.0)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (l <= 2.1e+236) {
		tmp = d * (pow((h * l), -0.5) * ((-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * ((0.5d0 * (h * (((d_m / ((d * 2.0d0) / m_m)) ** 2.0d0) / l))) + (-1.0d0))
    else if (l <= 2.1d+236) then
        tmp = d * (((h * l) ** (-0.5d0)) * (((-0.5d0) * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)) + 1.0d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * ((0.5 * (h * (Math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (l <= 2.1e+236) {
		tmp = d * (Math.pow((h * l), -0.5) * ((-0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * ((0.5 * (h * (math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0)
	elif l <= 2.1e+236:
		tmp = d * (math.pow((h * l), -0.5) * ((-0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0));
	elseif (l <= 2.1e+236)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)) + 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (h * (((D_m / ((d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0);
	elseif (l <= 2.1e+236)
		tmp = d * (((h * l) ^ -0.5) * ((-0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l)) + 1.0));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.1e+236], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.2%

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

   3. Simplified 66.6%

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

      1. clear-num 66.6%

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

      2. un-div-inv 66.6%

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

      3. frac-times 68.2%

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

      4. associate-/l* 66.6%

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

      5. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-/r/ 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 76.3%

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

       7. neg-mul-1 76.3%

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

   11. Simplified 76.3%

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

   if -3.999999999999988e-310 < l < 2.10000000000000006e236

   1. Initial program 66.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in d around 0 72.0%

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

      1. *-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. pow1 72.0%

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

      2. pow1/2 72.0%

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

      3. inv-pow 72.0%

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

      4. pow-pow 72.6%

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

      5. metadata-eval 72.6%

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

      6. cancel-sign-sub-inv 72.6%

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

      7. metadata-eval 72.6%

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

      8. *-commutative 72.6%

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

      9. div-inv 72.6%

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

      10. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. unpow1 72.6%

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

       2. associate-*l* 74.4%

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

       3. *-commutative 74.4%

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

   11. Simplified 74.4%

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

       1. associate-*l/ 81.6%

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

       2. associate-*l* 81.6%

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

   13. Applied egg-rr 81.6%

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

   if 2.10000000000000006e236 < l

   1. Initial program 36.6%

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

   3. Simplified 36.3%

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

   6. Applied egg-rr 45.4%

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

   8. Simplified 45.8%

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

   9. Taylor expanded in d around inf 37.5%

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

   10. Taylor expanded in d around 0 46.9%

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

       1. unpow1/2 46.9%

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

       2. rem-exp-log 43.5%

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

       3. exp-neg 43.5%

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

       4. exp-prod 43.5%

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

       5. distribute-lft-neg-out 43.5%

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

       6. distribute-rgt-neg-in 43.5%

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

       7. metadata-eval 43.5%

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

       8. exp-to-pow 46.8%

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

   12. Simplified 46.8%

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

       1. *-commutative 46.8%

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

       2. unpow-prod-down 65.7%

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

   14. Applied egg-rr 65.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot t\_0\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{\frac{d \cdot 2}{M\_m}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -4e-310)
     (*
      (* d t_0)
      (+ (* 0.5 (* h (/ (pow (/ D_m (/ (* d 2.0) M_m)) 2.0) l))) -1.0))
     (if (<= l 2.75e+236)
       (*
        d
        (*
         t_0
         (+ (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)) 1.0)))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * t_0) * ((0.5 * (h * (pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (l <= 2.75e+236) {
		tmp = d * (t_0 * ((-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-4d-310)) then
        tmp = (d * t_0) * ((0.5d0 * (h * (((d_m / ((d * 2.0d0) / m_m)) ** 2.0d0) / l))) + (-1.0d0))
    else if (l <= 2.75d+236) then
        tmp = d * (t_0 * (((-0.5d0) * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)) + 1.0d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * t_0) * ((0.5 * (h * (Math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0);
	} else if (l <= 2.75e+236) {
		tmp = d * (t_0 * ((-0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -4e-310:
		tmp = (d * t_0) * ((0.5 * (h * (math.pow((D_m / ((d * 2.0) / M_m)), 2.0) / l))) + -1.0)
	elif l <= 2.75e+236:
		tmp = d * (t_0 * ((-0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(d * t_0) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m / Float64(Float64(d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0));
	elseif (l <= 2.75e+236)
		tmp = Float64(d * Float64(t_0 * Float64(Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)) + 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (d * t_0) * ((0.5 * (h * (((D_m / ((d * 2.0) / M_m)) ^ 2.0) / l))) + -1.0);
	elseif (l <= 2.75e+236)
		tmp = d * (t_0 * ((-0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l)) + 1.0));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(d * t$95$0), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.75e+236], N[(d * N[(t$95$0 * N[(N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.2%

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

   3. Simplified 66.6%

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

      1. clear-num 66.6%

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

      2. un-div-inv 66.6%

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

      3. frac-times 68.2%

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

      4. associate-/l* 66.6%

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

      5. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-/r/ 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 76.3%

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

       7. neg-mul-1 76.3%

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

   11. Simplified 76.3%

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

   12. Taylor expanded in l around 0 76.0%

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

       1. unpow1/2 76.0%

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

       2. rem-exp-log 73.6%

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

       3. exp-neg 73.6%

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

       4. exp-prod 73.6%

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

       5. distribute-lft-neg-out 73.6%

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

       6. distribute-rgt-neg-in 73.6%

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

       7. metadata-eval 73.6%

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

       8. exp-to-pow 76.0%

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

   14. Simplified 76.0%

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

   if -3.999999999999988e-310 < l < 2.75e236

   1. Initial program 66.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in d around 0 72.0%

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

      1. *-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. pow1 72.0%

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

      2. pow1/2 72.0%

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

      3. inv-pow 72.0%

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

      4. pow-pow 72.6%

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

      5. metadata-eval 72.6%

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

      6. cancel-sign-sub-inv 72.6%

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

      7. metadata-eval 72.6%

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

      8. *-commutative 72.6%

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

      9. div-inv 72.6%

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

      10. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. unpow1 72.6%

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

       2. associate-*l* 74.4%

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

       3. *-commutative 74.4%

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

   11. Simplified 74.4%

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

       1. associate-*l/ 81.6%

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

       2. associate-*l* 81.6%

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

   13. Applied egg-rr 81.6%

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

   if 2.75e236 < l

   1. Initial program 36.6%

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

   3. Simplified 36.3%

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

   6. Applied egg-rr 45.4%

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

   8. Simplified 45.8%

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

   9. Taylor expanded in d around inf 37.5%

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

   10. Taylor expanded in d around 0 46.9%

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

       1. unpow1/2 46.9%

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

       2. rem-exp-log 43.5%

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

       3. exp-neg 43.5%

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

       4. exp-prod 43.5%

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

       5. distribute-lft-neg-out 43.5%

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

       6. distribute-rgt-neg-in 43.5%

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

       7. metadata-eval 43.5%

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

       8. exp-to-pow 46.8%

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

   12. Simplified 46.8%

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

       1. *-commutative 46.8%

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

       2. unpow-prod-down 65.7%

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

   14. Applied egg-rr 65.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot t\_0\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -4e-310)
     (*
      (* d t_0)
      (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
     (if (<= l 1.45e+236)
       (*
        d
        (*
         t_0
         (+ (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)) 1.0)))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * t_0) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 1.45e+236) {
		tmp = d * (t_0 * ((-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-4d-310)) then
        tmp = (d * t_0) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 1.45d+236) then
        tmp = d * (t_0 * (((-0.5d0) * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)) + 1.0d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -4e-310) {
		tmp = (d * t_0) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 1.45e+236) {
		tmp = d * (t_0 * ((-0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -4e-310:
		tmp = (d * t_0) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	elif l <= 1.45e+236:
		tmp = d * (t_0 * ((-0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(d * t_0) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	elseif (l <= 1.45e+236)
		tmp = Float64(d * Float64(t_0 * Float64(Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)) + 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (d * t_0) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	elseif (l <= 1.45e+236)
		tmp = d * (t_0 * ((-0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l)) + 1.0));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(d * t$95$0), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+236], N[(d * N[(t$95$0 * N[(N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.2%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in d around 0 2.3%

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

      1. *-commutative 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 72.1%

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

      4. mul-1-neg 72.1%

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

      5. associate-/l/ 72.5%

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

      6. unpow1/2 72.5%

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

      7. associate-/r* 72.1%

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

      8. rem-exp-log 70.0%

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

      9. exp-neg 70.0%

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

      10. exp-prod 70.0%

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

      11. distribute-lft-neg-out 70.0%

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

      12. distribute-rgt-neg-in 70.0%

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

      13. metadata-eval 70.0%

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

      14. exp-to-pow 72.1%

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

      15. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   if -3.999999999999988e-310 < l < 1.45e236

   1. Initial program 66.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in d around 0 72.0%

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

      1. *-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. pow1 72.0%

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

      2. pow1/2 72.0%

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

      3. inv-pow 72.0%

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

      4. pow-pow 72.6%

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

      5. metadata-eval 72.6%

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

      6. cancel-sign-sub-inv 72.6%

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

      7. metadata-eval 72.6%

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

      8. *-commutative 72.6%

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

      9. div-inv 72.6%

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

      10. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. unpow1 72.6%

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

       2. associate-*l* 74.4%

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

       3. *-commutative 74.4%

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

   11. Simplified 74.4%

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

       1. associate-*l/ 81.6%

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

       2. associate-*l* 81.6%

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

   13. Applied egg-rr 81.6%

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

   if 1.45e236 < l

   1. Initial program 36.6%

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

   3. Simplified 36.3%

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

   6. Applied egg-rr 45.4%

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

   8. Simplified 45.8%

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

   9. Taylor expanded in d around inf 37.5%

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

   10. Taylor expanded in d around 0 46.9%

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

       1. unpow1/2 46.9%

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

       2. rem-exp-log 43.5%

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

       3. exp-neg 43.5%

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

       4. exp-prod 43.5%

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

       5. distribute-lft-neg-out 43.5%

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

       6. distribute-rgt-neg-in 43.5%

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

       7. metadata-eval 43.5%

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

       8. exp-to-pow 46.8%

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

   12. Simplified 46.8%

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

       1. *-commutative 46.8%

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

       2. unpow-prod-down 65.7%

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

   14. Applied egg-rr 65.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-42}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \left(-D\_m\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.1e-42)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= d -2e-310)
     (*
      -0.125
      (/ (* (sqrt (/ h (pow l 3.0))) (* (* M_m D_m) (* M_m (- D_m)))) d))
     (*
      d
      (*
       (pow (* h l) -0.5)
       (+ (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)) 1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.1e-42) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -2e-310) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (pow((h * l), -0.5) * ((-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.1d-42)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-2d-310)) then
        tmp = (-0.125d0) * ((sqrt((h / (l ** 3.0d0))) * ((m_m * d_m) * (m_m * -d_m))) / d)
    else
        tmp = d * (((h * l) ** (-0.5d0)) * (((-0.5d0) * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)) + 1.0d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.1e-42) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -2e-310) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (Math.pow((h * l), -0.5) * ((-0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.1e-42:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -2e-310:
		tmp = -0.125 * ((math.sqrt((h / math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d)
	else:
		tmp = d * (math.pow((h * l), -0.5) * ((-0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)) + 1.0))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.1e-42)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -2e-310)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(M_m * D_m) * Float64(M_m * Float64(-D_m)))) / d));
	else
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)) + 1.0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.1e-42)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -2e-310)
		tmp = -0.125 * ((sqrt((h / (l ^ 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	else
		tmp = d * (((h * l) ^ -0.5) * ((-0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.1e-42], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * (-D$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10000000000000003e-42

   1. Initial program 81.1%

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

   3. Simplified 79.8%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in d around inf 0.0%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 56.8%

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

       4. associate-*r* 56.8%

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

       5. *-commutative 56.8%

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

       6. neg-mul-1 56.8%

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

       7. *-commutative 56.8%

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

       8. associate-/r* 57.3%

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

   12. Simplified 57.3%

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

   if -1.10000000000000003e-42 < d < -1.999999999999994e-310

   1. Initial program 46.3%

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

   3. Simplified 40.1%

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

      1. clear-num 40.1%

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

      2. sqrt-div 40.0%

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

      3. metadata-eval 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. *-un-lft-identity 40.0%

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

      2. pow1/2 40.0%

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

      3. pow-flip 40.1%

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

      4. metadata-eval 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. *-lft-identity 40.1%

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

   10. Simplified 40.1%

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

   11. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l/ 0.0%

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

       2. associate-*r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 38.3%

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

       5. associate-*l* 38.3%

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

       6. unpow2 38.3%

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

       7. unpow2 38.3%

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

       8. swap-sqr 51.7%

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

       9. unpow2 51.7%

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

       10. mul-1-neg 51.7%

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

   13. Simplified 51.7%

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

       1. unpow2 51.7%

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

       2. *-commutative 51.7%

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

       3. *-commutative 51.7%

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

   15. Applied egg-rr 51.7%

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

   if -1.999999999999994e-310 < d

   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 60.8%

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

   5. Taylor expanded in d around 0 66.5%

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

      1. *-commutative 66.5%

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

   7. Simplified 66.5%

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

      1. pow1 66.5%

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

      2. pow1/2 66.5%

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

      3. inv-pow 66.5%

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

      4. pow-pow 67.0%

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

      5. metadata-eval 67.0%

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

      6. cancel-sign-sub-inv 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. div-inv 67.0%

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

      10. metadata-eval 67.0%

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

   9. Applied egg-rr 67.0%

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

       1. unpow1 67.0%

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

       2. associate-*l* 68.6%

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

       3. *-commutative 68.6%

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

   11. Simplified 68.6%

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

       1. associate-*l/ 74.7%

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

       2. associate-*l* 74.7%

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

   13. Applied egg-rr 74.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-42}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot \left(-D\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-42}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \left(-D\_m\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \left(h \cdot \frac{{\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right) + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.1e-42)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= d -2e-310)
     (*
      -0.125
      (/ (* (sqrt (/ h (pow l 3.0))) (* (* M_m D_m) (* M_m (- D_m)))) d))
     (*
      d
      (*
       (pow (* h l) -0.5)
       (+ (* -0.5 (* h (/ (pow (* (* M_m 0.5) (/ D_m d)) 2.0) l))) 1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.1e-42) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -2e-310) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (pow((h * l), -0.5) * ((-0.5 * (h * (pow(((M_m * 0.5) * (D_m / d)), 2.0) / l))) + 1.0));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.1d-42)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-2d-310)) then
        tmp = (-0.125d0) * ((sqrt((h / (l ** 3.0d0))) * ((m_m * d_m) * (m_m * -d_m))) / d)
    else
        tmp = d * (((h * l) ** (-0.5d0)) * (((-0.5d0) * (h * ((((m_m * 0.5d0) * (d_m / d)) ** 2.0d0) / l))) + 1.0d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.1e-42) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -2e-310) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (Math.pow((h * l), -0.5) * ((-0.5 * (h * (Math.pow(((M_m * 0.5) * (D_m / d)), 2.0) / l))) + 1.0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.1e-42:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -2e-310:
		tmp = -0.125 * ((math.sqrt((h / math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d)
	else:
		tmp = d * (math.pow((h * l), -0.5) * ((-0.5 * (h * (math.pow(((M_m * 0.5) * (D_m / d)), 2.0) / l))) + 1.0))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.1e-42)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -2e-310)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(M_m * D_m) * Float64(M_m * Float64(-D_m)))) / d));
	else
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(Float64(-0.5 * Float64(h * Float64((Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0) / l))) + 1.0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.1e-42)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -2e-310)
		tmp = -0.125 * ((sqrt((h / (l ^ 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	else
		tmp = d * (((h * l) ^ -0.5) * ((-0.5 * (h * ((((M_m * 0.5) * (D_m / d)) ^ 2.0) / l))) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.1e-42], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * (-D$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(-0.5 * N[(h * N[(N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10000000000000003e-42

   1. Initial program 81.1%

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

   3. Simplified 79.8%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in d around inf 0.0%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 56.8%

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

       4. associate-*r* 56.8%

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

       5. *-commutative 56.8%

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

       6. neg-mul-1 56.8%

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

       7. *-commutative 56.8%

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

       8. associate-/r* 57.3%

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

   12. Simplified 57.3%

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

   if -1.10000000000000003e-42 < d < -1.999999999999994e-310

   1. Initial program 46.3%

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

   3. Simplified 40.1%

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

      1. clear-num 40.1%

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

      2. sqrt-div 40.0%

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

      3. metadata-eval 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. *-un-lft-identity 40.0%

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

      2. pow1/2 40.0%

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

      3. pow-flip 40.1%

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

      4. metadata-eval 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. *-lft-identity 40.1%

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

   10. Simplified 40.1%

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

   11. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l/ 0.0%

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

       2. associate-*r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 38.3%

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

       5. associate-*l* 38.3%

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

       6. unpow2 38.3%

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

       7. unpow2 38.3%

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

       8. swap-sqr 51.7%

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

       9. unpow2 51.7%

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

       10. mul-1-neg 51.7%

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

   13. Simplified 51.7%

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

       1. unpow2 51.7%

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

       2. *-commutative 51.7%

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

       3. *-commutative 51.7%

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

   15. Applied egg-rr 51.7%

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

   if -1.999999999999994e-310 < d

   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 60.8%

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

   5. Taylor expanded in d around 0 66.5%

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

      1. *-commutative 66.5%

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

   7. Simplified 66.5%

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

      1. pow1 66.5%

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

      2. pow1/2 66.5%

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

      3. inv-pow 66.5%

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

      4. pow-pow 67.0%

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

      5. metadata-eval 67.0%

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

      6. cancel-sign-sub-inv 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. div-inv 67.0%

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

      10. metadata-eval 67.0%

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

   9. Applied egg-rr 67.0%

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

       1. unpow1 67.0%

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

       2. associate-*l* 68.6%

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

       3. *-commutative 68.6%

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

   11. Simplified 68.6%

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

       1. associate-*l/ 74.7%

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

       2. associate-*l* 74.7%

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

   13. Applied egg-rr 74.7%

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

       1. associate-/l* 74.7%

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

       2. associate-*r* 74.7%

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

       3. *-commutative 74.7%

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

   15. Simplified 74.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-42}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot \left(-D\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-0.5 \cdot \left(h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-49}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-285}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \left(-D\_m\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.8e-49)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= d 1.05e-285)
     (*
      -0.125
      (/ (* (sqrt (/ h (pow l 3.0))) (* (* M_m D_m) (* M_m (- D_m)))) d))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.8e-49) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= 1.05e-285) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.8d-49)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= 1.05d-285) then
        tmp = (-0.125d0) * ((sqrt((h / (l ** 3.0d0))) * ((m_m * d_m) * (m_m * -d_m))) / d)
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.8e-49) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= 1.05e-285) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.8e-49:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= 1.05e-285:
		tmp = -0.125 * ((math.sqrt((h / math.pow(l, 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.8e-49)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= 1.05e-285)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(M_m * D_m) * Float64(M_m * Float64(-D_m)))) / d));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.8e-49)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= 1.05e-285)
		tmp = -0.125 * ((sqrt((h / (l ^ 3.0))) * ((M_m * D_m) * (M_m * -D_m))) / d);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.8e-49], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-285], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * (-D$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.8e-49

   1. Initial program 81.1%

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

   3. Simplified 79.8%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in d around inf 0.0%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 56.8%

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

       4. associate-*r* 56.8%

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

       5. *-commutative 56.8%

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

       6. neg-mul-1 56.8%

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

       7. *-commutative 56.8%

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

       8. associate-/r* 57.3%

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

   12. Simplified 57.3%

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

   if -5.8e-49 < d < 1.04999999999999992e-285

   1. Initial program 42.8%

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

   3. Simplified 37.0%

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

      1. clear-num 37.0%

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

      2. sqrt-div 36.9%

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

      3. metadata-eval 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. *-un-lft-identity 36.9%

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

      2. pow1/2 36.9%

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

      3. pow-flip 37.0%

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

      4. metadata-eval 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. *-lft-identity 37.0%

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

   10. Simplified 37.0%

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

   11. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l/ 0.0%

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

       2. associate-*r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 35.4%

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

       5. associate-*l* 35.4%

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

       6. unpow2 35.4%

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

       7. unpow2 35.4%

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

       8. swap-sqr 47.8%

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

       9. unpow2 47.8%

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

       10. mul-1-neg 47.8%

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

   13. Simplified 47.8%

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

       1. unpow2 47.8%

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

       2. *-commutative 47.8%

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

       3. *-commutative 47.8%

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

   15. Applied egg-rr 47.8%

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

   if 1.04999999999999992e-285 < d

   1. Initial program 63.5%

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

   3. Simplified 62.6%

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

   6. Applied egg-rr 70.8%

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

   8. Simplified 73.5%

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

   9. Taylor expanded in d around inf 40.1%

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

   10. Taylor expanded in d around 0 43.1%

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

       1. unpow1/2 43.1%

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

       2. rem-exp-log 41.0%

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

       3. exp-neg 41.0%

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

       4. exp-prod 41.5%

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

       5. distribute-lft-neg-out 41.5%

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

       6. distribute-rgt-neg-in 41.5%

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

       7. metadata-eval 41.5%

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

       8. exp-to-pow 43.6%

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

   12. Simplified 43.6%

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

       1. *-commutative 43.6%

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

       2. unpow-prod-down 51.3%

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

   14. Applied egg-rr 51.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-49}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-285}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot \left(-D\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-213}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 5.8e-213)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.8e-213) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 5.8d-213) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.8e-213) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 5.8e-213:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 5.8e-213)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 5.8e-213)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 5.8e-213], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.7999999999999999e-213

   1. Initial program 68.8%

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

   3. Simplified 67.4%

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

   6. Applied egg-rr 8.0%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in d around inf 2.2%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 43.9%

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

       4. associate-*r* 43.9%

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

       5. *-commutative 43.9%

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

       6. neg-mul-1 43.9%

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

       7. *-commutative 43.9%

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

       8. associate-/r* 44.2%

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

   12. Simplified 44.2%

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

   if 5.7999999999999999e-213 < l

   1. Initial program 60.1%

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

   3. Simplified 59.2%

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

   6. Applied egg-rr 68.1%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in d around inf 41.3%

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

   10. Taylor expanded in d around 0 44.6%

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

       1. unpow1/2 44.6%

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

       2. rem-exp-log 42.3%

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

       3. exp-neg 42.3%

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

       4. exp-prod 42.9%

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

       5. distribute-lft-neg-out 42.9%

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

       6. distribute-rgt-neg-in 42.9%

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

       7. metadata-eval 42.9%

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

       8. exp-to-pow 45.2%

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

   12. Simplified 45.2%

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

       1. *-commutative 45.2%

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

       2. unpow-prod-down 53.6%

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

   14. Applied egg-rr 53.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-213}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-213}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 5.8e-213) (* (- d) t_0) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= 5.8e-213) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (l <= 5.8d-213) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= 5.8e-213) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= 5.8e-213:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= 5.8e-213)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= 5.8e-213)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 5.8e-213], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.7999999999999999e-213

   1. Initial program 68.8%

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

   3. Simplified 67.4%

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

   6. Applied egg-rr 8.0%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in d around inf 2.2%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 43.9%

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

       4. associate-*r* 43.9%

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

       5. *-commutative 43.9%

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

       6. neg-mul-1 43.9%

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

       7. *-commutative 43.9%

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

       8. associate-/r* 44.2%

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

   12. Simplified 44.2%

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

   if 5.7999999999999999e-213 < l

   1. Initial program 60.1%

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

   3. Simplified 59.2%

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

   6. Applied egg-rr 68.1%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in d around inf 41.3%

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

   10. Taylor expanded in d around 0 44.6%

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

       1. associate-/r* 45.2%

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

   12. Simplified 45.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-213}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-213}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 6.5e-213)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 6.5e-213) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 6.5d-213) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 6.5e-213) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 6.5e-213:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 6.5e-213)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 6.5e-213)
		tmp = d * -sqrt((1.0 / (h * l)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 6.5e-213], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5e-213

   1. Initial program 68.8%

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

   3. Simplified 67.4%

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

   6. Applied egg-rr 8.0%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in d around inf 2.2%

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

   10. Taylor expanded in d around 0 11.5%

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

       1. unpow1/2 11.5%

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

       2. rem-exp-log 11.4%

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

       3. exp-neg 11.4%

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

       4. exp-prod 11.4%

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

       5. distribute-lft-neg-out 11.4%

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

       6. distribute-rgt-neg-in 11.4%

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

       7. metadata-eval 11.4%

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

       8. exp-to-pow 11.5%

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

   12. Simplified 11.5%

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

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   14. 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%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 43.9%

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

       6. neg-mul-1 43.9%

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

   15. Simplified 43.9%

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

   if 6.5e-213 < l

   1. Initial program 60.1%

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

   3. Simplified 59.2%

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

   6. Applied egg-rr 68.1%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in d around inf 41.3%

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

   10. Taylor expanded in d around 0 44.6%

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

       1. associate-/r* 45.2%

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

   12. Simplified 45.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-213}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-210}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.15e-210)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.15e-210) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 1.15d-210) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.15e-210) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1.15e-210:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.15e-210)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1.15e-210)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.15e-210], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-210

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 9.4%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 2.2%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-*l* 0.0%

          \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 43.9%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. associate-*r* 43.9%

          \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       5. *-commutative 43.9%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. neg-mul-1 43.9%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       7. *-commutative 43.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

       8. unpow1/2 43.9%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-d\right) \]

       9. rem-exp-log 42.1%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-d\right) \]

       10. exp-neg 42.1%

           \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-d\right) \]

       11. exp-prod 42.1%

           \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-d\right) \]

       12. distribute-lft-neg-out 42.1%

           \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-d\right) \]

       13. distribute-rgt-neg-in 42.1%

           \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 42.1%

           \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-d\right) \]

       15. exp-to-pow 43.9%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-d\right) \]

   12. Simplified 43.9%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.15e-210 < l

   1. Initial program 60.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 69.3%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 41.3%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]

   10. Taylor expanded in d around 0 44.6%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-/r* 45.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   12. Simplified 45.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-210}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.8e-190) (sqrt (* d (/ (/ d h) l))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.8e-190) {
		tmp = sqrt((d * ((d / h) / l)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.8d-190)) then
        tmp = sqrt((d * ((d / h) / l)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.8e-190) {
		tmp = Math.sqrt((d * ((d / h) / l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.8e-190:
		tmp = math.sqrt((d * ((d / h) / l)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.8e-190)
		tmp = sqrt(Float64(d * Float64(Float64(d / h) / l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.8e-190)
		tmp = sqrt((d * ((d / h) / l)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.8e-190], N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.80000000000000005e-190

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 0.0%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 0.0%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. sqrt-unprod 37.5%

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

       2. associate-/r* 37.4%

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{\frac{d}{h}}{\ell}}} \]

   11. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}} \]

   if -2.80000000000000005e-190 < d

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 32.7%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]

   10. Taylor expanded in d around 0 38.1%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-/r* 38.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.7e-190) (sqrt (* d (/ (/ d h) l))) (* d (pow (* h l) -0.5))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.7e-190) {
		tmp = sqrt((d * ((d / h) / l)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.7d-190)) then
        tmp = sqrt((d * ((d / h) / l)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.7e-190) {
		tmp = Math.sqrt((d * ((d / h) / l)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.7e-190:
		tmp = math.sqrt((d * ((d / h) / l)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.7e-190)
		tmp = sqrt(Float64(d * Float64(Float64(d / h) / l)));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.7e-190)
		tmp = sqrt((d * ((d / h) / l)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.7e-190], N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.6999999999999999e-190

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 0.0%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 0.0%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. sqrt-unprod 37.5%

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

       2. associate-/r* 37.4%

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{\frac{d}{h}}{\ell}}} \]

   11. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}} \]

   if -2.6999999999999999e-190 < d

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

   9. Taylor expanded in d around inf 32.7%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]

   10. Taylor expanded in d around 0 38.1%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. unpow1/2 38.1%

          \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

       2. rem-exp-log 36.4%

          \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       3. exp-neg 36.4%

          \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       4. exp-prod 36.9%

          \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       5. distribute-lft-neg-out 36.9%

          \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       6. distribute-rgt-neg-in 36.9%

          \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

       7. metadata-eval 36.9%

          \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

       8. exp-to-pow 38.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (pow (* h l) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * pow((h * l), -0.5);
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * ((h * l) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.pow((h * l), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.pow((h * l), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

6. Applied egg-rr 35.4%

   \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}} \]

8. Simplified 36.8%

   \[\leadsto \color{blue}{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{h}}\right)} \]

9. Taylor expanded in d around inf 20.1%

   \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]

10. Taylor expanded in d around 0 26.6%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. unpow1/2 26.6%

       \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

    2. rem-exp-log 25.5%

       \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

    3. exp-neg 25.5%

       \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

    4. exp-prod 25.8%

       \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

    5. distribute-lft-neg-out 25.8%

       \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

    6. distribute-rgt-neg-in 25.8%

       \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

    7. metadata-eval 25.8%

       \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

    8. exp-to-pow 26.9%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

12. Simplified 26.9%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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)))))