Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 79.2%

Time: 32.3s

Alternatives: 28

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := M\_m \cdot \frac{D\_m}{d \cdot 2}\\ \mathbf{if}\;h \leq -1.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {t\_1}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(t\_1 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (- d))) (t_1 (* M_m (/ D_m (* d 2.0)))))
   (if (<= h -1.4e+64)
     (*
      (* (/ t_0 (sqrt (- h))) (/ 1.0 (sqrt (/ l d))))
      (- 1.0 (* 0.5 (/ (* h (pow t_1 2.0)) l))))
     (if (<= h -1.05e-306)
       (*
        (* (sqrt (/ d h)) (/ t_0 (sqrt (- l))))
        (- 1.0 (* 0.5 (pow (* t_1 (sqrt (/ h l))) 2.0))))
       (*
        d
        (/
         (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 t_0 = sqrt(-d);
	double t_1 = M_m * (D_m / (d * 2.0));
	double tmp;
	if (h <= -1.4e+64) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * pow(t_1, 2.0)) / l)));
	} else if (h <= -1.05e-306) {
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * pow((t_1 * sqrt((h / l))), 2.0)));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(M_m * Float64(D_m / Float64(d * 2.0)))
	tmp = 0.0
	if (h <= -1.4e+64)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (t_1 ^ 2.0)) / l))));
	elseif (h <= -1.05e-306)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * (Float64(t_1 * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.4e+64], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1.05e-306], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$1 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $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 < -1.40000000000000012e64

   1. Initial program 63.6%

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

   3. Simplified 61.5%

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

      1. associate-*r/ 63.9%

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

      2. frac-times 66.0%

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

      3. associate-/l* 63.9%

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

      4. *-commutative 63.9%

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

   6. Applied egg-rr 63.9%

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

      1. frac-2neg 63.9%

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

      2. sqrt-div 78.7%

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

   8. Applied egg-rr 78.7%

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

      1. clear-num 78.8%

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

      2. sqrt-div 78.9%

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

      3. metadata-eval 78.9%

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

   10. Applied egg-rr 78.9%

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

   if -1.40000000000000012e64 < h < -1.0500000000000001e-306

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

      1. add-sqr-sqrt 78.4%

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

      2. pow2 78.4%

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

      3. sqrt-prod 78.5%

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

      4. sqrt-pow1 81.7%

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

      5. metadata-eval 81.7%

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

      6. frac-times 80.2%

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

      7. associate-/l* 81.7%

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

      8. pow1 81.7%

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

      9. *-commutative 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. frac-2neg 81.7%

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

      2. sqrt-div 90.3%

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

   8. Applied egg-rr 90.3%

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

   if -1.0500000000000001e-306 < h

   1. Initial program 70.0%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 81.6%

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

      1. unpow1 81.6%

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

      2. associate-*l/ 85.0%

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

      3. associate-/l* 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-*r* 84.9%

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

      6. fma-define 84.9%

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

      7. *-commutative 84.9%

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

      8. associate-*r/ 86.3%

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

      9. *-commutative 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{h}}{\ell}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M\_m \cdot D\_m\right)}^{2} \cdot \left(\sqrt{t\_0} \cdot \left(-\left|t\_0\right|\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (cbrt h) l)))
   (if (<= d -3.8e-155)
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D_m (* d 2.0))) 2.0)) l)))
      (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (if (<= d 1.4e-307)
       (* -0.125 (/ (* (pow (* M_m D_m) 2.0) (* (sqrt t_0) (- (fabs t_0)))) d))
       (*
        d
        (/
         (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 t_0 = cbrt(h) / l;
	double tmp;
	if (d <= -3.8e-155) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (D_m / (d * 2.0))), 2.0)) / l))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else if (d <= 1.4e-307) {
		tmp = -0.125 * ((pow((M_m * D_m), 2.0) * (sqrt(t_0) * -fabs(t_0))) / d);
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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)
	t_0 = Float64(cbrt(h) / l)
	tmp = 0.0
	if (d <= -3.8e-155)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (d <= 1.4e-307)
		tmp = Float64(-0.125 * Float64(Float64((Float64(M_m * D_m) ^ 2.0) * Float64(sqrt(t_0) * Float64(-abs(t_0)))) / d));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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_] := Block[{t$95$0 = N[(N[Power[h, 1/3], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -3.8e-155], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-307], N[(-0.125 * N[(N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[Abs[t$95$0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $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 < -3.7999999999999998e-155

   1. Initial program 81.3%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 80.3%

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

      2. frac-times 81.6%

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

      3. associate-/l* 80.3%

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

      4. *-commutative 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. frac-2neg 80.3%

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

      2. sqrt-div 86.4%

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

   8. Applied egg-rr 86.4%

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

   if -3.7999999999999998e-155 < d < 1.4e-307

   1. Initial program 45.8%

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

   3. Simplified 45.8%

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

      1. associate-*r/ 45.9%

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

      2. frac-times 45.9%

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

      3. associate-/l* 45.9%

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

      4. *-commutative 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. *-commutative 45.9%

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

      2. associate-/l* 43.2%

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

      3. associate-*r/ 43.2%

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

      4. *-commutative 43.2%

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

   8. Simplified 43.2%

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

   9. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. 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 40.2%

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

       5. associate-*l* 40.2%

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

       6. unpow2 40.2%

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

       7. unpow2 40.2%

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

       8. swap-sqr 49.8%

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

       9. unpow2 49.8%

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

       10. mul-1-neg 49.8%

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

   11. Simplified 49.8%

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

       1. pow1/2 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\color{blue}{{\left(\frac{h}{{\ell}^{3}}\right)}^{0.5}}\right)}{d} \]

       2. add-cube-cbrt 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\color{blue}{\left(\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right) \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}}^{0.5}\right)}{d} \]

       3. unpow-prod-down 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\color{blue}{{\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}}\right)}{d} \]

       4. pow2 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\color{blue}{\left({\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{2}\right)}}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)}{d} \]

       5. cbrt-div 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)}{d} \]

       6. unpow3 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)}{d} \]

       7. add-cbrt-cube 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)}{d} \]

       8. cbrt-div 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{0.5}\right)}{d} \]

       9. unpow3 49.8%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{0.5}\right)}{d} \]

       10. add-cbrt-cube 64.7%

           \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-{\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{0.5}\right)}{d} \]

   13. Applied egg-rr 64.7%

       \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\color{blue}{{\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}}\right)}{d} \]
   14. Step-by-step derivation

       1. unpow1/2 64.7%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\color{blue}{\sqrt{{\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right)}{d} \]

       2. unpow2 64.7%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\sqrt{\color{blue}{\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right)}{d} \]

       3. rem-sqrt-square 71.6%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\color{blue}{\left|\frac{\sqrt[3]{h}}{\ell}\right|} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right)}{d} \]

       4. unpow1/2 71.6%

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

   15. Simplified 71.6%

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

   if 1.4e-307 < d

   1. Initial program 71.0%

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

   3. Simplified 69.6%

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

   6. Applied egg-rr 82.7%

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

      1. unpow1 82.7%

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

      2. associate-*l/ 86.2%

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

      3. associate-/l* 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-*r* 86.1%

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

      6. fma-define 86.1%

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

      7. *-commutative 86.1%

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

      8. associate-*r/ 87.5%

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

      9. *-commutative 87.5%

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

   8. Simplified 87.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot \left(\sqrt{\frac{\sqrt[3]{h}}{\ell}} \cdot \left(-\left|\frac{\sqrt[3]{h}}{\ell}\right|\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -7.5e+18)
     (*
      (* (/ t_0 (sqrt (- h))) (/ 1.0 (sqrt (/ l d))))
      (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D_m (* d 2.0))) 2.0)) l))))
     (if (<= h -1.05e-306)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))))))
       (*
        d
        (/
         (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 t_0 = sqrt(-d);
	double tmp;
	if (h <= -7.5e+18) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * pow((M_m * (D_m / (d * 2.0))), 2.0)) / l)));
	} else if (h <= -1.05e-306) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -7.5e+18)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0)) / l))));
	elseif (h <= -1.05e-306)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -7.5e+18], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1.05e-306], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $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 < -7.5e18

   1. Initial program 67.0%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-*r/ 67.3%

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

      2. frac-times 69.1%

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

      3. associate-/l* 67.3%

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

      4. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. frac-2neg 67.3%

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

      2. sqrt-div 79.9%

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

   8. Applied egg-rr 79.9%

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

      1. clear-num 80.0%

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

      2. sqrt-div 80.0%

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

      3. metadata-eval 80.0%

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

   10. Applied egg-rr 80.0%

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

   if -7.5e18 < h < -1.0500000000000001e-306

   1. Initial program 77.1%

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

   3. Simplified 75.5%

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

      1. frac-2neg 75.5%

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

      2. sqrt-div 83.8%

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

   6. Applied egg-rr 83.8%

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

   if -1.0500000000000001e-306 < h

   1. Initial program 70.0%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 81.6%

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

      1. unpow1 81.6%

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

      2. associate-*l/ 85.0%

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

      3. associate-/l* 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-*r* 84.9%

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

      6. fma-define 84.9%

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

      7. *-commutative 84.9%

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

      8. associate-*r/ 86.3%

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

      9. *-commutative 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -8.2e+18)
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D_m (* d 2.0))) 2.0)) l)))
      (* (/ t_0 (sqrt (- h))) (sqrt (/ d l))))
     (if (<= h -1.05e-306)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))))))
       (*
        d
        (/
         (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 t_0 = sqrt(-d);
	double tmp;
	if (h <= -8.2e+18) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (D_m / (d * 2.0))), 2.0)) / l))) * ((t_0 / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= -1.05e-306) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -8.2e+18)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= -1.05e-306)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -8.2e+18], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1.05e-306], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $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 < -8.2e18

   1. Initial program 67.0%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-*r/ 67.3%

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

      2. frac-times 69.1%

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

      3. associate-/l* 67.3%

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

      4. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. frac-2neg 67.3%

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

      2. sqrt-div 79.9%

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

   8. Applied egg-rr 79.9%

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

   if -8.2e18 < h < -1.0500000000000001e-306

   1. Initial program 77.1%

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

   3. Simplified 75.5%

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

      1. frac-2neg 75.5%

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

      2. sqrt-div 83.8%

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

   6. Applied egg-rr 83.8%

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

   if -1.0500000000000001e-306 < h

   1. Initial program 70.0%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 81.6%

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

      1. unpow1 81.6%

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

      2. associate-*l/ 85.0%

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

      3. associate-/l* 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-*r* 84.9%

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

      6. fma-define 84.9%

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

      7. *-commutative 84.9%

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

      8. associate-*r/ 86.3%

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

      9. *-commutative 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1e-309)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D_m (* d 2.0))) 2.0)) l)))
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
   (*
    d
    (/
     (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 <= -1e-309) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (D_m / (d * 2.0))), 2.0)) / l))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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 <= -1e-309)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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, -1e-309], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 71.4%

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

   3. Simplified 70.5%

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

      1. associate-*r/ 70.7%

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

      2. frac-times 71.6%

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

      3. associate-/l* 70.7%

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

      4. *-commutative 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. frac-2neg 70.7%

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

      2. sqrt-div 78.7%

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

   8. Applied egg-rr 78.7%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

   6. Applied egg-rr 82.2%

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

      1. unpow1 82.2%

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

      2. associate-*l/ 85.6%

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

      3. associate-/l* 85.5%

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

      4. +-commutative 85.5%

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

      5. associate-*r* 85.5%

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

      6. fma-define 85.5%

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

      7. *-commutative 85.5%

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

      8. associate-*r/ 86.9%

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

      9. *-commutative 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1e-309)
   (*
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (*
    d
    (/
     (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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 <= -1e-309) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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 <= -1e-309)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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, -1e-309], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 71.4%

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

   3. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 78.4%

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

   6. Applied egg-rr 78.4%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

   6. Applied egg-rr 82.2%

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

      1. unpow1 82.2%

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

      2. associate-*l/ 85.6%

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

      3. associate-/l* 85.5%

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

      4. +-commutative 85.5%

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

      5. associate-*r* 85.5%

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

      6. fma-define 85.5%

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

      7. *-commutative 85.5%

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

      8. associate-*r/ 86.9%

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

      9. *-commutative 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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-83)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D_m (* d 2.0))) 2.0)) l)))
    (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))))
   (*
    d
    (/
     (fma (* (/ h l) -0.5) (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 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-83) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (D_m / (d * 2.0))), 2.0)) / l))) * (sqrt((d / l)) * (1.0 / sqrt((h / d))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m * D_m) / (d * 2.0)), 2.0), 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-83)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0), 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-83], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.50000000000000005e-83

   1. Initial program 73.8%

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

   3. Simplified 73.2%

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

      1. associate-*r/ 75.4%

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

      2. frac-times 76.0%

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

      3. associate-/l* 75.4%

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

      4. *-commutative 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. clear-num 75.4%

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

      2. sqrt-div 75.4%

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

      3. metadata-eval 75.4%

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

   8. Applied egg-rr 75.4%

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

   if 1.50000000000000005e-83 < l

   1. Initial program 66.4%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 81.3%

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

      1. unpow1 81.3%

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

      2. associate-*l/ 86.2%

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

      3. associate-/l* 87.0%

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

      4. +-commutative 87.0%

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

      5. associate-*r* 87.0%

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

      6. fma-define 87.0%

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

      7. *-commutative 87.0%

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

      8. associate-*r/ 88.9%

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

      9. *-commutative 88.9%

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

   8. Simplified 88.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(0.125 \cdot \frac{{\left(M\_m \cdot D\_m\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1.55e-176)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0) l)))))
   (if (<= d 1.4e-307)
     (* (sqrt (* h (pow l -3.0))) (* 0.125 (/ (pow (* M_m D_m) 2.0) d)))
     (*
      d
      (/
       (+ 1.0 (* (pow (* M_m (/ D_m (* d 2.0))) 2.0) (* (/ h l) -0.5)))
       (* (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 <= -1.55e-176) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))));
	} else if (d <= 1.4e-307) {
		tmp = sqrt((h * pow(l, -3.0))) * (0.125 * (pow((M_m * D_m), 2.0) / d));
	} else {
		tmp = d * ((1.0 + (pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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) :: tmp
    if (d <= (-1.55d-176)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * (h * ((((d / m_m) * (2.0d0 / d_m)) ** (-2.0d0)) / l))))
    else if (d <= 1.4d-307) then
        tmp = sqrt((h * (l ** (-3.0d0)))) * (0.125d0 * (((m_m * d_m) ** 2.0d0) / d))
    else
        tmp = d * ((1.0d0 + (((m_m * (d_m / (d * 2.0d0))) ** 2.0d0) * ((h / l) * (-0.5d0)))) / (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 tmp;
	if (d <= -1.55e-176) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (Math.pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))));
	} else if (d <= 1.4e-307) {
		tmp = Math.sqrt((h * Math.pow(l, -3.0))) * (0.125 * (Math.pow((M_m * D_m), 2.0) / d));
	} else {
		tmp = d * ((1.0 + (Math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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):
	tmp = 0
	if d <= -1.55e-176:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (math.pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))))
	elif d <= 1.4e-307:
		tmp = math.sqrt((h * math.pow(l, -3.0))) * (0.125 * (math.pow((M_m * D_m), 2.0) / d))
	else:
		tmp = d * ((1.0 + (math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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)
	tmp = 0.0
	if (d <= -1.55e-176)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0) / l)))));
	elseif (d <= 1.4e-307)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(0.125 * Float64((Float64(M_m * D_m) ^ 2.0) / d)));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64((Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0) * Float64(Float64(h / l) * -0.5))) / 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)
	tmp = 0.0;
	if (d <= -1.55e-176)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * ((((d / M_m) * (2.0 / D_m)) ^ -2.0) / l))));
	elseif (d <= 1.4e-307)
		tmp = sqrt((h * (l ^ -3.0))) * (0.125 * (((M_m * D_m) ^ 2.0) / d));
	else
		tmp = d * ((1.0 + (((M_m * (D_m / (d * 2.0))) ^ 2.0) * ((h / l) * -0.5))) / (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_] := If[LessEqual[d, -1.55e-176], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-307], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.54999999999999996e-176

   1. Initial program 78.8%

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

   3. Simplified 77.7%

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

      1. associate-*r/ 78.0%

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

      2. frac-times 79.1%

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

      3. associate-/l* 78.0%

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

      4. *-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. pow1 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. unpow1 78.0%

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

      2. *-commutative 78.0%

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

      3. unpow2 78.0%

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

      4. *-lft-identity 78.0%

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

      5. associate-*r/ 78.0%

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

      6. associate-*r/ 78.0%

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

      7. associate-*l/ 78.0%

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

      8. associate-/r/ 78.0%

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

      9. unpow-1 78.0%

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

      10. *-lft-identity 78.0%

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

      11. associate-*r/ 79.2%

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

      12. associate-*r/ 79.2%

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

      13. associate-*l/ 79.1%

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

      14. associate-/r/ 79.1%

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

      15. unpow-1 79.1%

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

      16. pow-sqr 79.2%

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

      17. times-frac 78.1%

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

      18. metadata-eval 78.1%

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

   10. Simplified 78.1%

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

       1. pow1 78.1%

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

       2. sqrt-unprod 73.5%

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

       3. cancel-sign-sub-inv 73.5%

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

       4. metadata-eval 73.5%

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

       5. associate-/l* 74.7%

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

   12. Applied egg-rr 74.7%

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

       1. unpow1 74.7%

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

       2. *-commutative 74.7%

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

   14. Simplified 74.7%

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

   if -1.54999999999999996e-176 < d < 1.4e-307

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. associate-*r/ 46.7%

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

      2. frac-times 46.7%

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

      3. associate-/l* 46.7%

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

      4. *-commutative 46.7%

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

   6. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. associate-/l* 46.7%

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

      3. associate-*r/ 46.7%

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

      4. *-commutative 46.7%

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

   8. Simplified 46.7%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. associate-*r* 0.0%

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

       2. *-commutative 0.0%

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

       3. *-commutative 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 47.2%

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

       7. associate-*l* 47.2%

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

       8. *-commutative 47.2%

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

       9. mul-1-neg 47.2%

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

       10. unpow2 47.2%

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

       11. unpow2 47.2%

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

       12. swap-sqr 54.5%

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

       13. unpow2 54.5%

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

   11. Simplified 54.5%

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

       1. pow1 54.5%

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

       2. associate-*r* 54.5%

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

       3. div-inv 54.5%

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

       4. pow-flip 58.0%

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

       5. metadata-eval 58.0%

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

       6. *-commutative 58.0%

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

   13. Applied egg-rr 58.0%

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

       1. unpow1 58.0%

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

       2. associate-*l* 58.0%

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

       3. distribute-frac-neg 58.0%

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

       4. neg-mul-1 58.0%

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

       5. associate-*r* 58.0%

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

       6. metadata-eval 58.0%

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

       7. unpow2 58.0%

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

       8. swap-sqr 50.8%

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

       9. unpow2 50.8%

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

       10. unpow2 50.8%

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

       11. *-commutative 50.8%

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

       12. *-commutative 50.8%

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

       13. unpow2 50.8%

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

       14. unpow2 50.8%

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

       15. swap-sqr 58.0%

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

       16. unpow2 58.0%

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

       17. *-commutative 58.0%

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

   15. Simplified 58.0%

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

   if 1.4e-307 < d

   1. Initial program 71.0%

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

   3. Simplified 69.6%

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

   6. Applied egg-rr 82.7%

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

      1. unpow1 82.7%

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

      2. associate-*l/ 86.2%

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

      3. associate-/l* 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-*r* 86.1%

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

      6. fma-define 86.1%

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

      7. *-commutative 86.1%

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

      8. associate-*r/ 87.5%

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

      9. *-commutative 87.5%

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

   8. Simplified 87.5%

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

      1. fma-undefine 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r/ 86.1%

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

   10. Applied egg-rr 86.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot t\_0}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (* M_m (/ D_m (* d 2.0))) 2.0)))
   (if (<= l 1e-83)
     (*
      (- 1.0 (* 0.5 (/ (* h t_0) l)))
      (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))))
     (* d (/ (+ 1.0 (* t_0 (* (/ h l) -0.5))) (* (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 = pow((M_m * (D_m / (d * 2.0))), 2.0);
	double tmp;
	if (l <= 1e-83) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (sqrt((d / l)) * (1.0 / sqrt((h / d))));
	} else {
		tmp = d * ((1.0 + (t_0 * ((h / l) * -0.5))) / (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 = (m_m * (d_m / (d * 2.0d0))) ** 2.0d0
    if (l <= 1d-83) then
        tmp = (1.0d0 - (0.5d0 * ((h * t_0) / l))) * (sqrt((d / l)) * (1.0d0 / sqrt((h / d))))
    else
        tmp = d * ((1.0d0 + (t_0 * ((h / l) * (-0.5d0)))) / (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 = Math.pow((M_m * (D_m / (d * 2.0))), 2.0);
	double tmp;
	if (l <= 1e-83) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d))));
	} else {
		tmp = d * ((1.0 + (t_0 * ((h / l) * -0.5))) / (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 = math.pow((M_m * (D_m / (d * 2.0))), 2.0)
	tmp = 0
	if l <= 1e-83:
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (math.sqrt((d / l)) * (1.0 / math.sqrt((h / d))))
	else:
		tmp = d * ((1.0 + (t_0 * ((h / l) * -0.5))) / (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(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0
	tmp = 0.0
	if (l <= 1e-83)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(t_0 * Float64(Float64(h / l) * -0.5))) / 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 = (M_m * (D_m / (d * 2.0))) ^ 2.0;
	tmp = 0.0;
	if (l <= 1e-83)
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (sqrt((d / l)) * (1.0 / sqrt((h / d))));
	else
		tmp = d * ((1.0 + (t_0 * ((h / l) * -0.5))) / (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[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 1e-83], N[(N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1e-83

   1. Initial program 73.8%

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

   3. Simplified 73.2%

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

      1. associate-*r/ 75.4%

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

      2. frac-times 76.0%

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

      3. associate-/l* 75.4%

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

      4. *-commutative 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. clear-num 75.4%

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

      2. sqrt-div 75.4%

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

      3. metadata-eval 75.4%

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

   8. Applied egg-rr 75.4%

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

   if 1e-83 < l

   1. Initial program 66.4%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 81.3%

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

      1. unpow1 81.3%

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

      2. associate-*l/ 86.2%

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

      3. associate-/l* 87.0%

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

      4. +-commutative 87.0%

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

      5. associate-*r* 87.0%

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

      6. fma-define 87.0%

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

      7. *-commutative 87.0%

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

      8. associate-*r/ 88.9%

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

      9. *-commutative 88.9%

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

   8. Simplified 88.9%

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

      1. fma-undefine 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*r/ 87.0%

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

   10. Applied egg-rr 87.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D\_m \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 4.2e-120)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0) l)))))
   (if (<= l 2.7e+99)
     (/
      (* d (fma (pow (* D_m (/ M_m (* d 2.0))) 2.0) (* (/ h l) -0.5) 1.0))
      (sqrt (* h l)))
     (/ (/ 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 tmp;
	if (l <= 4.2e-120) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))));
	} else if (l <= 2.7e+99) {
		tmp = (d * fma(pow((D_m * (M_m / (d * 2.0))), 2.0), ((h / l) * -0.5), 1.0)) / sqrt((h * l));
	} else {
		tmp = (d / 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 <= 4.2e-120)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0) / l)))));
	elseif (l <= 2.7e+99)
		tmp = Float64(Float64(d * fma((Float64(D_m * Float64(M_m / Float64(d * 2.0))) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0)) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(d / 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, 4.2e-120], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e+99], N[(N[(d * N[(N[Power[N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.2000000000000001e-120

   1. Initial program 73.3%

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

   3. Simplified 72.7%

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

      1. associate-*r/ 74.9%

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

      2. frac-times 75.6%

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

      3. associate-/l* 74.9%

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

      4. *-commutative 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. pow1 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. unpow1 74.9%

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

      2. *-commutative 74.9%

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

      3. unpow2 74.9%

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

      4. *-lft-identity 74.9%

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

      5. associate-*r/ 74.9%

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

      6. associate-*r/ 74.9%

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

      7. associate-*l/ 74.9%

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

      8. associate-/r/ 74.9%

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

      9. unpow-1 74.9%

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

      10. *-lft-identity 74.9%

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

      11. associate-*r/ 75.6%

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

      12. associate-*r/ 75.6%

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

      13. associate-*l/ 75.6%

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

      14. associate-/r/ 75.6%

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

      15. unpow-1 75.6%

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

      16. pow-sqr 75.6%

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

      17. times-frac 75.0%

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

      18. metadata-eval 75.0%

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

   10. Simplified 75.0%

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

       1. pow1 75.0%

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

       2. sqrt-unprod 70.3%

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

       3. cancel-sign-sub-inv 70.3%

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

       4. metadata-eval 70.3%

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

       5. associate-/l* 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. unpow1 70.3%

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

       2. *-commutative 70.3%

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

   14. Simplified 70.3%

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

   if 4.2000000000000001e-120 < l < 2.69999999999999989e99

   1. Initial program 70.6%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 87.2%

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

      1. unpow1 87.2%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*r/ 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-*r/ 88.9%

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

      7. associate-*r* 88.9%

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

      8. associate-*r* 88.9%

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

      9. associate-/r* 88.9%

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

   8. Simplified 88.9%

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

      1. associate-*l/ 92.5%

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

      2. +-commutative 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-/l/ 92.5%

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

      5. *-commutative 92.5%

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

      6. associate-/l* 92.5%

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

      7. fma-define 92.5%

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

      8. associate-/l* 92.5%

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

      9. sqrt-unprod 84.2%

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

      10. *-commutative 84.2%

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

   10. Applied egg-rr 84.2%

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

   if 2.69999999999999989e99 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}\\ \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+99}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot t\_0, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0)))
   (if (<= l 2.1e-120)
     (* (sqrt (* (/ d h) (/ d l))) (+ 1.0 (* -0.5 (* h (/ t_0 l)))))
     (if (<= l 2.55e+99)
       (* d (/ (fma (/ h l) (* -0.5 t_0) 1.0) (sqrt (* h l))))
       (/ (/ 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 = pow(((d / M_m) * (2.0 / D_m)), -2.0);
	double tmp;
	if (l <= 2.1e-120) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (t_0 / l))));
	} else if (l <= 2.55e+99) {
		tmp = d * (fma((h / l), (-0.5 * t_0), 1.0) / sqrt((h * l)));
	} else {
		tmp = (d / 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)
	t_0 = Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0
	tmp = 0.0
	if (l <= 2.1e-120)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64(t_0 / l)))));
	elseif (l <= 2.55e+99)
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * t_0), 1.0) / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(d / 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_] := Block[{t$95$0 = N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[l, 2.1e-120], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.55e+99], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.1e-120

   1. Initial program 73.3%

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

   3. Simplified 72.7%

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

      1. associate-*r/ 74.9%

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

      2. frac-times 75.6%

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

      3. associate-/l* 74.9%

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

      4. *-commutative 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. pow1 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. unpow1 74.9%

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

      2. *-commutative 74.9%

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

      3. unpow2 74.9%

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

      4. *-lft-identity 74.9%

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

      5. associate-*r/ 74.9%

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

      6. associate-*r/ 74.9%

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

      7. associate-*l/ 74.9%

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

      8. associate-/r/ 74.9%

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

      9. unpow-1 74.9%

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

      10. *-lft-identity 74.9%

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

      11. associate-*r/ 75.6%

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

      12. associate-*r/ 75.6%

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

      13. associate-*l/ 75.6%

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

      14. associate-/r/ 75.6%

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

      15. unpow-1 75.6%

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

      16. pow-sqr 75.6%

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

      17. times-frac 75.0%

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

      18. metadata-eval 75.0%

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

   10. Simplified 75.0%

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

       1. pow1 75.0%

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

       2. sqrt-unprod 70.3%

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

       3. cancel-sign-sub-inv 70.3%

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

       4. metadata-eval 70.3%

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

       5. associate-/l* 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. unpow1 70.3%

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

       2. *-commutative 70.3%

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

   14. Simplified 70.3%

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

   if 2.1e-120 < l < 2.54999999999999976e99

   1. Initial program 70.6%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 87.2%

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

      1. unpow1 87.2%

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

      2. associate-*l/ 90.8%

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

      3. associate-/l* 90.8%

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

      4. +-commutative 90.8%

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

      5. associate-*r* 90.8%

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

      6. fma-define 90.8%

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

      7. *-commutative 90.8%

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

      8. associate-*r/ 92.5%

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

      9. *-commutative 92.5%

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

   8. Simplified 92.5%

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

      1. *-un-lft-identity 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*r/ 90.8%

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

      4. *-commutative 90.8%

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

      5. sqrt-prod 82.5%

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

   10. Applied egg-rr 82.5%

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

       1. *-lft-identity 82.5%

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

   12. Simplified 84.1%

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

   if 2.54999999999999976e99 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+99}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 1.7e-83)
   (*
    (- 1.0 (* 0.5 (* h (/ (pow (/ (* M_m D_m) (* d 2.0)) 2.0) l))))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (*
    d
    (/
     (+ 1.0 (* (pow (* M_m (/ D_m (* d 2.0))) 2.0) (* (/ h l) -0.5)))
     (* (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 <= 1.7e-83) {
		tmp = (1.0 - (0.5 * (h * (pow(((M_m * D_m) / (d * 2.0)), 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = d * ((1.0 + (pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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) :: tmp
    if (l <= 1.7d-83) then
        tmp = (1.0d0 - (0.5d0 * (h * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) / l)))) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = d * ((1.0d0 + (((m_m * (d_m / (d * 2.0d0))) ** 2.0d0) * ((h / l) * (-0.5d0)))) / (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 tmp;
	if (l <= 1.7e-83) {
		tmp = (1.0 - (0.5 * (h * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) / l)))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = d * ((1.0 + (Math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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):
	tmp = 0
	if l <= 1.7e-83:
		tmp = (1.0 - (0.5 * (h * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) / l)))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = d * ((1.0 + (math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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)
	tmp = 0.0
	if (l <= 1.7e-83)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) / l)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64((Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0) * Float64(Float64(h / l) * -0.5))) / 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)
	tmp = 0.0;
	if (l <= 1.7e-83)
		tmp = (1.0 - (0.5 * (h * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = d * ((1.0 + (((M_m * (D_m / (d * 2.0))) ^ 2.0) * ((h / l) * -0.5))) / (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_] := If[LessEqual[l, 1.7e-83], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6999999999999999e-83

   1. Initial program 73.8%

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

   3. Simplified 73.2%

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

      1. associate-*r/ 75.4%

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

      2. frac-times 76.0%

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

      3. associate-/l* 75.4%

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

      4. *-commutative 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-/l* 74.8%

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

      3. associate-*r/ 75.4%

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

      4. *-commutative 75.4%

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

   8. Simplified 75.4%

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

   if 1.6999999999999999e-83 < l

   1. Initial program 66.4%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 81.3%

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

      1. unpow1 81.3%

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

      2. associate-*l/ 86.2%

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

      3. associate-/l* 87.0%

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

      4. +-commutative 87.0%

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

      5. associate-*r* 87.0%

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

      6. fma-define 87.0%

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

      7. *-commutative 87.0%

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

      8. associate-*r/ 88.9%

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

      9. *-commutative 88.9%

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

   8. Simplified 88.9%

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

      1. fma-undefine 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*r/ 87.0%

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

   10. Applied egg-rr 87.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-83}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -5.8e-297)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))))))
   (*
    d
    (/
     (+ 1.0 (* (pow (* M_m (/ D_m (* d 2.0))) 2.0) (* (/ h l) -0.5)))
     (* (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 <= -5.8e-297) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else {
		tmp = d * ((1.0 + (pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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) :: tmp
    if (l <= (-5.8d-297)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0)))))
    else
        tmp = d * ((1.0d0 + (((m_m * (d_m / (d * 2.0d0))) ** 2.0d0) * ((h / l) * (-0.5d0)))) / (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 tmp;
	if (l <= -5.8e-297) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else {
		tmp = d * ((1.0 + (Math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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):
	tmp = 0
	if l <= -5.8e-297:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))))
	else:
		tmp = d * ((1.0 + (math.pow((M_m * (D_m / (d * 2.0))), 2.0) * ((h / l) * -0.5))) / (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)
	tmp = 0.0
	if (l <= -5.8e-297)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64((Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0) * Float64(Float64(h / l) * -0.5))) / 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)
	tmp = 0.0;
	if (l <= -5.8e-297)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0)))));
	else
		tmp = d * ((1.0 + (((M_m * (D_m / (d * 2.0))) ^ 2.0) * ((h / l) * -0.5))) / (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_] := If[LessEqual[l, -5.8e-297], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.79999999999999979e-297

   1. Initial program 72.7%

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

   3. Simplified 71.9%

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

   if -5.79999999999999979e-297 < l

   1. Initial program 69.5%

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

   3. Simplified 68.2%

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

   6. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-*l/ 84.4%

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

      3. associate-/l* 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-*r* 84.4%

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

      6. fma-define 84.4%

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

      7. *-commutative 84.4%

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

      8. associate-*r/ 85.7%

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

      9. *-commutative 85.7%

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

   8. Simplified 85.7%

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

      1. fma-undefine 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-*r/ 84.4%

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + t\_0 \cdot {\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2} \cdot t\_0}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (* (/ h l) -0.5)))
   (if (<= l -5.8e-297)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (+ 1.0 (* t_0 (pow (* (/ D_m d) (* 0.5 M_m)) 2.0)))))
     (*
      d
      (/
       (+ 1.0 (* (pow (* M_m (/ D_m (* d 2.0))) 2.0) t_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 t_0 = (h / l) * -0.5;
	double tmp;
	if (l <= -5.8e-297) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (t_0 * pow(((D_m / d) * (0.5 * M_m)), 2.0))));
	} else {
		tmp = d * ((1.0 + (pow((M_m * (D_m / (d * 2.0))), 2.0) * t_0)) / (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 = (h / l) * (-0.5d0)
    if (l <= (-5.8d-297)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (t_0 * (((d_m / d) * (0.5d0 * m_m)) ** 2.0d0))))
    else
        tmp = d * ((1.0d0 + (((m_m * (d_m / (d * 2.0d0))) ** 2.0d0) * t_0)) / (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 = (h / l) * -0.5;
	double tmp;
	if (l <= -5.8e-297) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (t_0 * Math.pow(((D_m / d) * (0.5 * M_m)), 2.0))));
	} else {
		tmp = d * ((1.0 + (Math.pow((M_m * (D_m / (d * 2.0))), 2.0) * t_0)) / (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 = (h / l) * -0.5
	tmp = 0
	if l <= -5.8e-297:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (t_0 * math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))
	else:
		tmp = d * ((1.0 + (math.pow((M_m * (D_m / (d * 2.0))), 2.0) * t_0)) / (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(Float64(h / l) * -0.5)
	tmp = 0.0
	if (l <= -5.8e-297)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(t_0 * (Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64((Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0) * t_0)) / 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 = (h / l) * -0.5;
	tmp = 0.0;
	if (l <= -5.8e-297)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (t_0 * (((D_m / d) * (0.5 * M_m)) ^ 2.0))));
	else
		tmp = d * ((1.0 + (((M_m * (D_m / (d * 2.0))) ^ 2.0) * t_0)) / (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[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[l, -5.8e-297], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.79999999999999979e-297

   1. Initial program 72.7%

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

   3. Simplified 71.0%

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

      1. fma-undefine 71.0%

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

      2. associate-*r* 71.0%

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

   6. Applied egg-rr 71.0%

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

   if -5.79999999999999979e-297 < l

   1. Initial program 69.5%

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

   3. Simplified 68.2%

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

   6. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-*l/ 84.4%

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

      3. associate-/l* 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-*r* 84.4%

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

      6. fma-define 84.4%

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

      7. *-commutative 84.4%

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

      8. associate-*r/ 85.7%

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

      9. *-commutative 85.7%

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

   8. Simplified 85.7%

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

      1. fma-undefine 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-*r/ 84.4%

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(h \cdot \ell + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -8.8e-100)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l -1e-309)
     (* d (exp (* 0.5 (- (log1p (+ (* h l) -1.0))))))
     (if (<= l 2.6e+99)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
       (/ (/ 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 tmp;
	if (l <= -8.8e-100) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d * exp((0.5 * -log1p(((h * l) + -1.0))));
	} else if (l <= 2.6e+99) {
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -8.8e-100) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d * Math.exp((0.5 * -Math.log1p(((h * l) + -1.0))));
	} else if (l <= 2.6e+99) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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):
	tmp = 0
	if l <= -8.8e-100:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= -1e-309:
		tmp = d * math.exp((0.5 * -math.log1p(((h * l) + -1.0))))
	elif l <= 2.6e+99:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= -8.8e-100)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -1e-309)
		tmp = Float64(d * exp(Float64(0.5 * Float64(-log1p(Float64(Float64(h * l) + -1.0))))));
	elseif (l <= 2.6e+99)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(d / 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, -8.8e-100], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(d * N[Exp[N[(0.5 * (-N[Log[1 + N[(N[(h * l), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+99], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.79999999999999957e-100

   1. Initial program 69.0%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in M around 0 51.3%

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

   if -8.79999999999999957e-100 < l < -1.000000000000002e-309

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in d around inf 25.7%

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

      1. pow1/2 25.7%

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

      2. pow-to-exp 25.7%

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

      3. log-rec 25.5%

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

   7. Applied egg-rr 25.5%

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

      1. log1p-expm1-u 57.3%

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

      2. expm1-undefine 57.3%

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

      3. add-exp-log 57.3%

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

   9. Applied egg-rr 57.3%

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

   if -1.000000000000002e-309 < l < 2.6e99

   1. Initial program 74.0%

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

   3. Simplified 73.0%

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

   6. Applied egg-rr 85.5%

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

      1. unpow1 85.5%

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

      2. associate-*r* 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-*r/ 86.5%

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

      5. *-commutative 86.5%

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

      6. associate-*r/ 86.5%

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

      7. associate-*r* 86.5%

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

      8. associate-*r* 86.5%

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

      9. associate-/r* 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in l around 0 75.0%

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

   if 2.6e99 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(h \cdot \ell + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 2.9e-120)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0) l)))))
   (if (<= l 4.7e+98)
     (*
      (/ d (sqrt (* h l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
     (/ (/ 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 tmp;
	if (l <= 2.9e-120) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))));
	} else if (l <= 4.7e+98) {
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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) :: tmp
    if (l <= 2.9d-120) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * (h * ((((d / m_m) * (2.0d0 / d_m)) ** (-2.0d0)) / l))))
    else if (l <= 4.7d+98) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    else
        tmp = (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 tmp;
	if (l <= 2.9e-120) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (Math.pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))));
	} else if (l <= 4.7e+98) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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):
	tmp = 0
	if l <= 2.9e-120:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (math.pow(((d / M_m) * (2.0 / D_m)), -2.0) / l))))
	elif l <= 4.7e+98:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= 2.9e-120)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0) / l)))));
	elseif (l <= 4.7e+98)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(d / 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)
	tmp = 0.0;
	if (l <= 2.9e-120)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * ((((d / M_m) * (2.0 / D_m)) ^ -2.0) / l))));
	elseif (l <= 4.7e+98)
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	else
		tmp = (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_] := If[LessEqual[l, 2.9e-120], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.7e+98], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.9e-120

   1. Initial program 73.3%

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

   3. Simplified 72.7%

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

      1. associate-*r/ 74.9%

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

      2. frac-times 75.6%

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

      3. associate-/l* 74.9%

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

      4. *-commutative 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. pow1 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. unpow1 74.9%

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

      2. *-commutative 74.9%

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

      3. unpow2 74.9%

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

      4. *-lft-identity 74.9%

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

      5. associate-*r/ 74.9%

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

      6. associate-*r/ 74.9%

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

      7. associate-*l/ 74.9%

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

      8. associate-/r/ 74.9%

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

      9. unpow-1 74.9%

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

      10. *-lft-identity 74.9%

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

      11. associate-*r/ 75.6%

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

      12. associate-*r/ 75.6%

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

      13. associate-*l/ 75.6%

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

      14. associate-/r/ 75.6%

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

      15. unpow-1 75.6%

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

      16. pow-sqr 75.6%

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

      17. times-frac 75.0%

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

      18. metadata-eval 75.0%

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

   10. Simplified 75.0%

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

       1. pow1 75.0%

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

       2. sqrt-unprod 70.3%

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

       3. cancel-sign-sub-inv 70.3%

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

       4. metadata-eval 70.3%

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

       5. associate-/l* 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. unpow1 70.3%

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

       2. *-commutative 70.3%

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

   14. Simplified 70.3%

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

   if 2.9e-120 < l < 4.6999999999999997e98

   1. Initial program 70.6%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 87.2%

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

      1. unpow1 87.2%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*r/ 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-*r/ 88.9%

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

      7. associate-*r* 88.9%

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

      8. associate-*r* 88.9%

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

      9. associate-/r* 88.9%

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

   8. Simplified 88.9%

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

   9. Taylor expanded in l around 0 80.6%

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

   if 4.6999999999999997e98 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M\_m \cdot \frac{D\_m}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 3.7e-140)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* (/ h l) (* (pow (* M_m (/ D_m (* d 2.0))) 2.0) -0.5))))
   (if (<= l 2.6e+99)
     (*
      (/ d (sqrt (* h l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
     (/ (/ 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 tmp;
	if (l <= 3.7e-140) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (pow((M_m * (D_m / (d * 2.0))), 2.0) * -0.5)));
	} else if (l <= 2.6e+99) {
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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) :: tmp
    if (l <= 3.7d-140) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((h / l) * (((m_m * (d_m / (d * 2.0d0))) ** 2.0d0) * (-0.5d0))))
    else if (l <= 2.6d+99) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    else
        tmp = (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 tmp;
	if (l <= 3.7e-140) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (Math.pow((M_m * (D_m / (d * 2.0))), 2.0) * -0.5)));
	} else if (l <= 2.6e+99) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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):
	tmp = 0
	if l <= 3.7e-140:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (math.pow((M_m * (D_m / (d * 2.0))), 2.0) * -0.5)))
	elif l <= 2.6e+99:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= 3.7e-140)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M_m * Float64(D_m / Float64(d * 2.0))) ^ 2.0) * -0.5))));
	elseif (l <= 2.6e+99)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(d / 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)
	tmp = 0.0;
	if (l <= 3.7e-140)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (((M_m * (D_m / (d * 2.0))) ^ 2.0) * -0.5)));
	elseif (l <= 2.6e+99)
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	else
		tmp = (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_] := If[LessEqual[l, 3.7e-140], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+99], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.69999999999999977e-140

   1. Initial program 74.3%

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

   3. Simplified 73.7%

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

      1. associate-*r/ 74.6%

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

      2. frac-times 75.2%

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

      3. associate-/l* 74.6%

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

      4. *-commutative 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. frac-2neg 74.6%

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

      2. sqrt-div 58.1%

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

   8. Applied egg-rr 58.1%

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

      1. pow1 58.1%

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

      2. sqrt-undiv 74.6%

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

      3. frac-2neg 74.6%

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

      4. sqrt-unprod 69.9%

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

      5. cancel-sign-sub-inv 69.9%

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

      6. metadata-eval 69.9%

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

      7. associate-/l* 69.0%

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

   10. Applied egg-rr 69.0%

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

       1. unpow1 69.0%

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

       2. associate-*r* 69.0%

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

       3. *-commutative 69.0%

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

   12. Simplified 69.0%

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

   if 3.69999999999999977e-140 < l < 2.6e99

   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 66.5%

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

   6. Applied egg-rr 84.2%

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

      1. unpow1 84.2%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*r/ 85.9%

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

      5. *-commutative 85.9%

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

      6. associate-*r/ 85.9%

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

      7. associate-*r* 85.9%

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

      8. associate-*r* 85.9%

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

      9. associate-/r* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in l around 0 77.9%

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

   if 2.6e99 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 3.5e-140)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* D_m (/ M_m (* d 2.0))) 2.0)))))
   (if (<= l 4.8e+98)
     (*
      (/ d (sqrt (* h l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
     (/ (/ 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 tmp;
	if (l <= 3.5e-140) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((D_m * (M_m / (d * 2.0))), 2.0))));
	} else if (l <= 4.8e+98) {
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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) :: tmp
    if (l <= 3.5d-140) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((d_m * (m_m / (d * 2.0d0))) ** 2.0d0))))
    else if (l <= 4.8d+98) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    else
        tmp = (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 tmp;
	if (l <= 3.5e-140) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((D_m * (M_m / (d * 2.0))), 2.0))));
	} else if (l <= 4.8e+98) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = (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):
	tmp = 0
	if l <= 3.5e-140:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((D_m * (M_m / (d * 2.0))), 2.0))))
	elif l <= 4.8e+98:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= 3.5e-140)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / Float64(d * 2.0))) ^ 2.0)))));
	elseif (l <= 4.8e+98)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(d / 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)
	tmp = 0.0;
	if (l <= 3.5e-140)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((D_m * (M_m / (d * 2.0))) ^ 2.0))));
	elseif (l <= 4.8e+98)
		tmp = (d / sqrt((h * l))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	else
		tmp = (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_] := If[LessEqual[l, 3.5e-140], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e+98], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.4999999999999998e-140

   1. Initial program 74.3%

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

   3. Simplified 73.7%

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

      1. associate-*r/ 74.6%

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

      2. frac-times 75.2%

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

      3. associate-/l* 74.6%

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

      4. *-commutative 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. pow1 74.6%

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

      2. sqrt-unprod 69.9%

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

      3. cancel-sign-sub-inv 69.9%

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

      4. metadata-eval 69.9%

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

      5. associate-/l* 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. unpow1 69.0%

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

      2. *-commutative 69.0%

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

      3. associate-*r/ 69.6%

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

      4. associate-*l/ 69.0%

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

      5. *-commutative 69.0%

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

      6. *-commutative 69.0%

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

   10. Simplified 69.0%

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

   if 3.4999999999999998e-140 < l < 4.7999999999999997e98

   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 66.5%

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

   6. Applied egg-rr 84.2%

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

      1. unpow1 84.2%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*r/ 85.9%

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

      5. *-commutative 85.9%

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

      6. associate-*r/ 85.9%

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

      7. associate-*r* 85.9%

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

      8. associate-*r* 85.9%

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

      9. associate-/r* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in l around 0 77.9%

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

   if 4.7999999999999997e98 < l

   1. Initial program 63.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in d around inf 42.1%

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

      1. sqrt-div 42.0%

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

      2. metadata-eval 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-unprod 63.4%

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

      5. div-inv 63.7%

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

      6. associate-/r* 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-275}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(h \cdot \ell + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -5.2e-100)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l 3.5e-275)
     (* d (exp (* 0.5 (- (log1p (+ (* h l) -1.0))))))
     (if (<= l 1.05e-198)
       (/ (- d) (sqrt (* h l)))
       (/ (/ 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 tmp;
	if (l <= -5.2e-100) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= 3.5e-275) {
		tmp = d * exp((0.5 * -log1p(((h * l) + -1.0))));
	} else if (l <= 1.05e-198) {
		tmp = -d / sqrt((h * l));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5.2e-100) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= 3.5e-275) {
		tmp = d * Math.exp((0.5 * -Math.log1p(((h * l) + -1.0))));
	} else if (l <= 1.05e-198) {
		tmp = -d / Math.sqrt((h * l));
	} else {
		tmp = (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):
	tmp = 0
	if l <= -5.2e-100:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= 3.5e-275:
		tmp = d * math.exp((0.5 * -math.log1p(((h * l) + -1.0))))
	elif l <= 1.05e-198:
		tmp = -d / math.sqrt((h * l))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= -5.2e-100)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= 3.5e-275)
		tmp = Float64(d * exp(Float64(0.5 * Float64(-log1p(Float64(Float64(h * l) + -1.0))))));
	elseif (l <= 1.05e-198)
		tmp = Float64(Float64(-d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(d / 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, -5.2e-100], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.5e-275], N[(d * N[Exp[N[(0.5 * (-N[Log[1 + N[(N[(h * l), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e-198], N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.1999999999999997e-100

   1. Initial program 69.0%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in M around 0 51.3%

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

   if -5.1999999999999997e-100 < l < 3.49999999999999969e-275

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in d around inf 28.8%

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

      1. pow1/2 28.8%

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

      2. pow-to-exp 28.5%

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

      3. log-rec 28.4%

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

   7. Applied egg-rr 28.4%

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

      1. log1p-expm1-u 52.7%

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

      2. expm1-undefine 52.7%

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

      3. add-exp-log 52.7%

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

   9. Applied egg-rr 52.7%

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

   if 3.49999999999999969e-275 < l < 1.04999999999999996e-198

   1. Initial program 87.3%

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

   3. Simplified 87.3%

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

      1. associate-*r/ 87.6%

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

      2. frac-times 87.6%

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

      3. associate-/l* 87.6%

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

      4. *-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-/l* 81.8%

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

      3. associate-*r/ 81.8%

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

      4. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in l around -inf 0.0%

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

       1. remove-double-neg 0.0%

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

       2. associate-*l* 0.0%

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

       3. *-commutative 0.0%

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

       4. distribute-rgt-neg-in 0.0%

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

       5. *-commutative 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 52.0%

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

       8. mul-1-neg 52.0%

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

       9. remove-double-neg 52.0%

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

       10. associate-/r* 52.0%

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

       11. unpow1/2 52.0%

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

       12. associate-/r* 52.0%

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

       13. rem-exp-log 52.0%

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

       14. exp-neg 52.0%

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

       15. exp-prod 52.0%

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

       16. distribute-lft-neg-out 52.0%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       17. exp-neg 52.0%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       18. exp-to-pow 52.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       19. unpow1/2 52.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. Simplified 52.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 1.04999999999999996e-198 < l

   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 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 42.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 42.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 42.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 56.7%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 56.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 56.9%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-275}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(h \cdot \ell + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{h}}{\ell}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{t\_0}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \sqrt[3]{{t\_0}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 (/ (/ 1.0 h) l)))
   (if (<= l -1.4e-115)
     (* (- d) (sqrt t_0))
     (if (<= l -5.8e-297)
       (* d (cbrt (pow t_0 1.5)))
       (if (<= l 1.3e-198)
         (/ (- d) (sqrt (* h l)))
         (/ (/ 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 = (1.0 / h) / l;
	double tmp;
	if (l <= -1.4e-115) {
		tmp = -d * sqrt(t_0);
	} else if (l <= -5.8e-297) {
		tmp = d * cbrt(pow(t_0, 1.5));
	} else if (l <= 1.3e-198) {
		tmp = -d / sqrt((h * l));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 / h) / l;
	double tmp;
	if (l <= -1.4e-115) {
		tmp = -d * Math.sqrt(t_0);
	} else if (l <= -5.8e-297) {
		tmp = d * Math.cbrt(Math.pow(t_0, 1.5));
	} else if (l <= 1.3e-198) {
		tmp = -d / Math.sqrt((h * l));
	} else {
		tmp = (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(Float64(1.0 / h) / l)
	tmp = 0.0
	if (l <= -1.4e-115)
		tmp = Float64(Float64(-d) * sqrt(t_0));
	elseif (l <= -5.8e-297)
		tmp = Float64(d * cbrt((t_0 ^ 1.5)));
	elseif (l <= 1.3e-198)
		tmp = Float64(Float64(-d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(d / 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_] := Block[{t$95$0 = N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, -1.4e-115], N[((-d) * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.8e-297], N[(d * N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e-198], N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.39999999999999994e-115

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 47.1%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 47.1%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 47.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.39999999999999994e-115 < l < -5.79999999999999979e-297

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 28.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 35.8%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 35.8%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 35.8%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 35.8%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 35.8%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 35.8%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/r* 35.8%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 35.8%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 35.8%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 35.8%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

   9. Simplified 35.8%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

   if -5.79999999999999979e-297 < l < 1.30000000000000003e-198

   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 76.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 76.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 73.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. remove-double-neg 0.0%

          \[\leadsto \color{blue}{-\left(-\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. associate-*l* 0.0%

          \[\leadsto -\left(-\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       3. *-commutative 0.0%

          \[\leadsto -\left(-d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

       4. distribute-rgt-neg-in 0.0%

          \[\leadsto -\color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       5. *-commutative 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

       6. unpow2 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       7. rem-square-sqrt 45.5%

          \[\leadsto -d \cdot \left(-\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       8. mul-1-neg 45.5%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       9. remove-double-neg 45.5%

          \[\leadsto -d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       10. associate-/r* 45.5%

           \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       11. unpow1/2 45.5%

           \[\leadsto -d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       12. associate-/r* 45.5%

           \[\leadsto -d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       13. rem-exp-log 45.2%

           \[\leadsto -d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       14. exp-neg 45.2%

           \[\leadsto -d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       15. exp-prod 45.2%

           \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       16. distribute-lft-neg-out 45.2%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       17. exp-neg 45.2%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       18. exp-to-pow 45.5%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       19. unpow1/2 45.5%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. Simplified 45.5%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 1.30000000000000003e-198 < l

   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 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 42.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 42.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 42.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 56.7%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 56.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 56.9%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-115}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -2.3e-115)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -5.8e-297)
     (/ d (cbrt (pow (* h l) 1.5)))
     (if (<= l 1.1e-198)
       (/ (- d) (sqrt (* h l)))
       (/ (/ 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 tmp;
	if (l <= -2.3e-115) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= -5.8e-297) {
		tmp = d / cbrt(pow((h * l), 1.5));
	} else if (l <= 1.1e-198) {
		tmp = -d / sqrt((h * l));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.3e-115) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -5.8e-297) {
		tmp = d / Math.cbrt(Math.pow((h * l), 1.5));
	} else if (l <= 1.1e-198) {
		tmp = -d / Math.sqrt((h * l));
	} else {
		tmp = (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)
	tmp = 0.0
	if (l <= -2.3e-115)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -5.8e-297)
		tmp = Float64(d / cbrt((Float64(h * l) ^ 1.5)));
	elseif (l <= 1.1e-198)
		tmp = Float64(Float64(-d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(d / 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, -2.3e-115], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.8e-297], N[(d / N[Power[N[Power[N[(h * l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e-198], N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.29999999999999985e-115

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 47.1%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 47.1%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 47.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -2.29999999999999985e-115 < l < -5.79999999999999979e-297

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 83.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 83.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 83.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 83.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 83.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 83.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 80.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 80.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 80.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 80.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in d around inf 28.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/r* 28.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       2. unpow1/2 28.9%

          \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       3. associate-/r* 28.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       4. rem-exp-log 28.9%

          \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       5. exp-neg 28.9%

          \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       6. exp-prod 28.8%

          \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 28.8%

          \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       8. exp-neg 28.8%

          \[\leadsto d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       9. exp-to-pow 28.8%

          \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       10. unpow1/2 28.8%

           \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       11. unpow-1 28.8%

           \[\leadsto d \cdot \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \]

       12. unpow-1 28.8%

           \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \]

       13. associate-/l* 28.8%

           \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       14. *-rgt-identity 28.8%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. add-cbrt-cube 35.6%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 35.6%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 35.6%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 35.6%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 35.6%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 35.6%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 35.6%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   13. Applied egg-rr 35.6%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   14. Step-by-step derivation

       1. unpow1/3 35.6%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   15. Simplified 35.6%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -5.79999999999999979e-297 < l < 1.1e-198

   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 76.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 76.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 73.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. remove-double-neg 0.0%

          \[\leadsto \color{blue}{-\left(-\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. associate-*l* 0.0%

          \[\leadsto -\left(-\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       3. *-commutative 0.0%

          \[\leadsto -\left(-d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

       4. distribute-rgt-neg-in 0.0%

          \[\leadsto -\color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       5. *-commutative 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

       6. unpow2 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       7. rem-square-sqrt 45.5%

          \[\leadsto -d \cdot \left(-\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       8. mul-1-neg 45.5%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       9. remove-double-neg 45.5%

          \[\leadsto -d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       10. associate-/r* 45.5%

           \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       11. unpow1/2 45.5%

           \[\leadsto -d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       12. associate-/r* 45.5%

           \[\leadsto -d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       13. rem-exp-log 45.2%

           \[\leadsto -d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       14. exp-neg 45.2%

           \[\leadsto -d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       15. exp-prod 45.2%

           \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       16. distribute-lft-neg-out 45.2%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       17. exp-neg 45.2%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       18. exp-to-pow 45.5%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       19. unpow1/2 45.5%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. Simplified 45.5%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 1.1e-198 < l

   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 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 42.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 42.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 42.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 56.7%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 56.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 56.9%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-115}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-297}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -1.9e+200)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l 1.45e-198)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (/ (/ 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 tmp;
	if (l <= -1.9e+200) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= 1.45e-198) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = (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) :: tmp
    if (l <= (-1.9d+200)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= 1.45d-198) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = (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 tmp;
	if (l <= -1.9e+200) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= 1.45e-198) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = (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):
	tmp = 0
	if l <= -1.9e+200:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= 1.45e-198:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = (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)
	tmp = 0.0
	if (l <= -1.9e+200)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= 1.45e-198)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(d / 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)
	tmp = 0.0;
	if (l <= -1.9e+200)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= 1.45e-198)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = (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_] := If[LessEqual[l, -1.9e+200], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e-198], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.89999999999999991e200

   1. Initial program 56.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 62.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -1.89999999999999991e200 < l < 1.45e-198

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 40.5%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 40.5%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 40.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 1.45e-198 < l

   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 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 42.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 42.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 42.2%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 56.7%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 56.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 56.9%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.1 \cdot 10^{-304}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (if (<= h 3.1e-304)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (/ 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 tmp;
	if (h <= 3.1e-304) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = 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) :: tmp
    if (h <= 3.1d-304) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = 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 tmp;
	if (h <= 3.1e-304) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = 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):
	tmp = 0
	if h <= 3.1e-304:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = 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)
	tmp = 0.0
	if (h <= 3.1e-304)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = 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)
	tmp = 0.0;
	if (h <= 3.1e-304)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = 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_] := If[LessEqual[h, 3.1e-304], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.09999999999999985e-304

   1. Initial program 71.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 39.2%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 39.2%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 39.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 3.09999999999999985e-304 < h

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 69.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 69.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 69.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 69.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 71.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in d around inf 38.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/r* 39.8%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       2. unpow1/2 39.8%

          \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       3. associate-/r* 38.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       4. rem-exp-log 37.1%

          \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       5. exp-neg 37.1%

          \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       6. exp-prod 37.8%

          \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 37.8%

          \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       8. exp-neg 37.9%

          \[\leadsto d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       9. exp-to-pow 39.6%

          \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       10. unpow1/2 39.6%

           \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       11. unpow-1 39.6%

           \[\leadsto d \cdot \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \]

       12. unpow-1 39.6%

           \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \]

       13. associate-/l* 39.7%

           \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       14. *-rgt-identity 39.7%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   11. Simplified 39.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. sqrt-prod 52.8%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       2. *-commutative 52.8%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   13. Applied egg-rr 52.8%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.1 \cdot 10^{-304}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 40.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (<= h 2.4e-270) (* (- 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 (h <= 2.4e-270) {
		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 (h <= 2.4d-270) 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 (h <= 2.4e-270) {
		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 h <= 2.4e-270:
		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 (h <= 2.4e-270)
		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 (h <= 2.4e-270)
		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[h, 2.4e-270], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 2.40000000000000002e-270

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 38.6%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 38.6%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 38.6%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 2.40000000000000002e-270 < h

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 41.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 41.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 40.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

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 (<= h 2.4e-270)
   (* (- 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 (h <= 2.4e-270) {
		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 (h <= 2.4d-270) 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 (h <= 2.4e-270) {
		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 h <= 2.4e-270:
		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 (h <= 2.4e-270)
		tmp = Float64(Float64(-d) * (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 (h <= 2.4e-270)
		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[h, 2.4e-270], 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 h < 2.40000000000000002e-270

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 71.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. pow1 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h\right)}^{1}}}{\ell}\right) \]

   8. Applied egg-rr 70.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h\right)}^{1}}}{\ell}\right) \]
   9. Step-by-step derivation

      1. unpow1 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}}{\ell}\right) \]

      2. *-commutative 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      3. unpow2 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}}{\ell}\right) \]

      4. *-lft-identity 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\color{blue}{\left(1 \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      5. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\left(1 \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      6. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\color{blue}{\frac{1 \cdot \left(M \cdot D\right)}{d \cdot 2}} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      7. associate-*l/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      8. associate-/r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M \cdot D}}} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      9. unpow-1 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(\color{blue}{{\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1}} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}{\ell}\right) \]

      10. *-lft-identity 70.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \color{blue}{\left(1 \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)\right)}\right)}{\ell}\right) \]

      11. associate-*r/ 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \left(1 \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right)\right)}{\ell}\right) \]

      12. associate-*r/ 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \color{blue}{\frac{1 \cdot \left(M \cdot D\right)}{d \cdot 2}}\right)}{\ell}\right) \]

      13. associate-*l/ 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}\right)}{\ell}\right) \]

      14. associate-/r/ 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{d \cdot 2}{M \cdot D}}}\right)}{\ell}\right) \]

      15. unpow-1 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left({\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1} \cdot \color{blue}{{\left(\frac{d \cdot 2}{M \cdot D}\right)}^{-1}}\right)}{\ell}\right) \]

      16. pow-sqr 71.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \color{blue}{{\left(\frac{d \cdot 2}{M \cdot D}\right)}^{\left(2 \cdot -1\right)}}}{\ell}\right) \]

      17. times-frac 70.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\color{blue}{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}}^{\left(2 \cdot -1\right)}}{\ell}\right) \]

      18. metadata-eval 70.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{-2}}}{\ell}\right) \]

   10. Simplified 70.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}}{\ell}\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 37.0%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative 37.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. neg-mul-1 37.0%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

       6. unpow-1 37.0%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 37.0%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 37.0%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 37.2%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. rem-square-sqrt 37.1%

           \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(-d\right) \]

       11. fabs-sqr 37.1%

           \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(-d\right) \]

       12. rem-square-sqrt 37.2%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-d\right) \]

   13. Simplified 37.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 2.40000000000000002e-270 < h

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 41.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 41.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 40.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

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 (<= h 2.4e-270) (/ (- d) (sqrt (* 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 (h <= 2.4e-270) {
		tmp = -d / sqrt((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 (h <= 2.4d-270) then
        tmp = -d / sqrt((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 (h <= 2.4e-270) {
		tmp = -d / Math.sqrt((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 h <= 2.4e-270:
		tmp = -d / math.sqrt((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 (h <= 2.4e-270)
		tmp = Float64(Float64(-d) / sqrt(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 (h <= 2.4e-270)
		tmp = -d / sqrt((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[h, 2.4e-270], N[((-d) / N[Sqrt[N[(h * 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 h < 2.40000000000000002e-270

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 71.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 71.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 71.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. remove-double-neg 0.0%

          \[\leadsto \color{blue}{-\left(-\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. associate-*l* 0.0%

          \[\leadsto -\left(-\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       3. *-commutative 0.0%

          \[\leadsto -\left(-d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

       4. distribute-rgt-neg-in 0.0%

          \[\leadsto -\color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       5. *-commutative 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

       6. unpow2 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       7. rem-square-sqrt 37.0%

          \[\leadsto -d \cdot \left(-\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       8. mul-1-neg 37.0%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       9. remove-double-neg 37.0%

          \[\leadsto -d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       10. associate-/r* 38.6%

           \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       11. unpow1/2 38.6%

           \[\leadsto -d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       12. associate-/r* 37.0%

           \[\leadsto -d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       13. rem-exp-log 35.3%

           \[\leadsto -d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       14. exp-neg 35.3%

           \[\leadsto -d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       15. exp-prod 35.5%

           \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       16. distribute-lft-neg-out 35.5%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       17. exp-neg 35.5%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       18. exp-to-pow 37.2%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       19. unpow1/2 37.2%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. Simplified 37.2%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 2.40000000000000002e-270 < h

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 41.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 41.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 41.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;d \leq -4 \cdot 10^{-174}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

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 (* h l)))) (if (<= d -4e-174) (/ (- 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((h * l));
	double tmp;
	if (d <= -4e-174) {
		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((h * l))
    if (d <= (-4d-174)) 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((h * l));
	double tmp;
	if (d <= -4e-174) {
		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((h * l))
	tmp = 0
	if d <= -4e-174:
		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(h * l))
	tmp = 0.0
	if (d <= -4e-174)
		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((h * l));
	tmp = 0.0;
	if (d <= -4e-174)
		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[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4e-174], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -4e-174

   1. Initial program 79.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 79.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 78.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 78.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 79.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 79.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 79.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. remove-double-neg 0.0%

          \[\leadsto \color{blue}{-\left(-\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. associate-*l* 0.0%

          \[\leadsto -\left(-\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       3. *-commutative 0.0%

          \[\leadsto -\left(-d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

       4. distribute-rgt-neg-in 0.0%

          \[\leadsto -\color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       5. *-commutative 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

       6. unpow2 0.0%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       7. rem-square-sqrt 41.6%

          \[\leadsto -d \cdot \left(-\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       8. mul-1-neg 41.6%

          \[\leadsto -d \cdot \left(-\color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

       9. remove-double-neg 41.6%

          \[\leadsto -d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       10. associate-/r* 43.9%

           \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       11. unpow1/2 43.9%

           \[\leadsto -d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       12. associate-/r* 41.6%

           \[\leadsto -d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       13. rem-exp-log 39.4%

           \[\leadsto -d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       14. exp-neg 39.4%

           \[\leadsto -d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       15. exp-prod 39.6%

           \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       16. distribute-lft-neg-out 39.6%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       17. exp-neg 39.6%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       18. exp-to-pow 41.9%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       19. unpow1/2 41.9%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. Simplified 41.9%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -4e-174 < d

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 66.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

   8. Simplified 68.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   9. Taylor expanded in d around inf 37.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/r* 38.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       2. unpow1/2 38.2%

          \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

       3. associate-/r* 37.5%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       4. rem-exp-log 36.0%

          \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

       5. exp-neg 36.0%

          \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

       6. exp-prod 36.6%

          \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 36.6%

          \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       8. exp-neg 36.6%

          \[\leadsto d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       9. exp-to-pow 38.0%

          \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       10. unpow1/2 38.0%

           \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       11. unpow-1 38.0%

           \[\leadsto d \cdot \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \]

       12. unpow-1 38.0%

           \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \]

       13. associate-/l* 38.1%

           \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       14. *-rgt-identity 38.1%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-174}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 26.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* 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) {
	return d / sqrt((h * l));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 69.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   2. frac-times 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

   3. associate-/l* 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

   4. *-commutative 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

6. Applied egg-rr 70.2%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
7. Step-by-step derivation

   1. *-commutative 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

   2. associate-/l* 70.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

   3. associate-*r/ 71.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

   4. *-commutative 71.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

8. Simplified 71.8%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

9. Taylor expanded in d around inf 28.0%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. associate-/r* 28.5%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

    2. unpow1/2 28.5%

       \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

    3. associate-/r* 28.0%

       \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

    4. rem-exp-log 27.0%

       \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

    5. exp-neg 27.0%

       \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

    6. exp-prod 27.4%

       \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

    7. distribute-lft-neg-out 27.4%

       \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

    8. exp-neg 27.4%

       \[\leadsto d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

    9. exp-to-pow 28.4%

       \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

    10. unpow1/2 28.4%

        \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

    11. unpow-1 28.4%

        \[\leadsto d \cdot \color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \]

    12. unpow-1 28.4%

        \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \]

    13. associate-/l* 28.4%

        \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

    14. *-rgt-identity 28.4%

        \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

11. Simplified 28.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(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)))))