Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 77.1%

Time: 17.7s

Alternatives: 23

Speedup: 3.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := 1 - \frac{h \cdot \frac{t\_0}{d}}{\ell} \cdot \frac{t\_0}{d \cdot 2}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot t\_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (* 0.5 M)))
        (t_1 (- 1.0 (* (/ (* h (/ t_0 d)) l) (/ t_0 (* d 2.0)))))
        (t_2 (sqrt (/ d h))))
   (if (<= d -1.5e-215)
     (* (* t_2 (/ (sqrt (- d)) (sqrt (- l)))) t_1)
     (if (<= d -5e-310)
       (fma
        (/ 1.0 (sqrt (/ (* l l) (/ h l))))
        (* (/ (* M M) d) (* 0.125 (* D D)))
        (* (- d) (sqrt (/ 1.0 (* h l)))))
       (* t_1 (* t_2 (/ (sqrt d) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (0.5 * M);
	double t_1 = 1.0 - (((h * (t_0 / d)) / l) * (t_0 / (d * 2.0)));
	double t_2 = sqrt((d / h));
	double tmp;
	if (d <= -1.5e-215) {
		tmp = (t_2 * (sqrt(-d) / sqrt(-l))) * t_1;
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(0.5 * M))
	t_1 = Float64(1.0 - Float64(Float64(Float64(h * Float64(t_0 / d)) / l) * Float64(t_0 / Float64(d * 2.0))))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -1.5e-215)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * t_1);
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(h * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.5e-215], N[(N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.50000000000000013e-215

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.2

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

   6. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 79.2

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

   8. Applied egg-rr 79.2%

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

      1. frac-2neg N/A

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

      2. lift-neg.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 89.4

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

   10. Applied egg-rr 89.4%

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

   if -1.50000000000000013e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 78.6

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

   6. Applied egg-rr 78.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 78.6

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (+ 1.0 (* (/ h l) (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
      -5e-109)
   (*
    (- 1.0 (/ (* (* h 0.5) (* M (* 0.25 (* D (* D M))))) (* d (* d l))))
    (sqrt (/ (* d d) (* h l))))
   (* (sqrt (/ d h)) (sqrt (/ d l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((D * M) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -5e-109) {
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 + ((h / l) * ((((d_1 * m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0))))) <= (-5d-109)) then
        tmp = (1.0d0 - (((h * 0.5d0) * (m * (0.25d0 * (d_1 * (d_1 * m))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)))
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((D * M) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -5e-109) {
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * Math.sqrt(((d * d) / (h * l)));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((D * M) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -5e-109:
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * math.sqrt(((d * d) / (h * l)))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))) <= -5e-109)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h * 0.5) * Float64(M * Float64(0.25 * Float64(D * Float64(D * M))))) / Float64(d * Float64(d * l)))) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((D * M) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))))) <= -5e-109)
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)));
	else
		tmp = sqrt((d / h)) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-109], N[(N[(1.0 - N[(N[(N[(h * 0.5), $MachinePrecision] * N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000002e-109

   1. Initial program 81.4%

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

   4. Applied egg-rr 79.2%

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

   6. Applied egg-rr 47.0%

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

   if -5.0000000000000002e-109 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.7%

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

   4. Applied egg-rr 53.3%

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

   5. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 33.5

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

   7. Simplified 33.5%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 33.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 33.8

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

   9. Applied egg-rr 33.8%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. pow1/2 N/A

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

       4. frac-times N/A

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

       5. sqrt-div N/A

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

       6. lift-/.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. pow1/2 N/A

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

       9. sqrt-div N/A

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

       10. lift-/.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 60.4

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

   11. Applied egg-rr 60.4%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := 1 - \frac{h \cdot \frac{t\_0}{d}}{\ell} \cdot \frac{t\_0}{d \cdot 2}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (* 0.5 M)))
        (t_1 (- 1.0 (* (/ (* h (/ t_0 d)) l) (/ t_0 (* d 2.0)))))
        (t_2 (sqrt (/ d h))))
   (if (<= d -1.5e-215)
     (* t_1 (* t_2 (sqrt (/ d l))))
     (if (<= d -5e-310)
       (fma
        (/ 1.0 (sqrt (/ (* l l) (/ h l))))
        (* (/ (* M M) d) (* 0.125 (* D D)))
        (* (- d) (sqrt (/ 1.0 (* h l)))))
       (* t_1 (* t_2 (/ (sqrt d) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (0.5 * M);
	double t_1 = 1.0 - (((h * (t_0 / d)) / l) * (t_0 / (d * 2.0)));
	double t_2 = sqrt((d / h));
	double tmp;
	if (d <= -1.5e-215) {
		tmp = t_1 * (t_2 * sqrt((d / l)));
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(0.5 * M))
	t_1 = Float64(1.0 - Float64(Float64(Float64(h * Float64(t_0 / d)) / l) * Float64(t_0 / Float64(d * 2.0))))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -1.5e-215)
		tmp = Float64(t_1 * Float64(t_2 * sqrt(Float64(d / l))));
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(h * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.5e-215], N[(t$95$1 * N[(t$95$2 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.50000000000000013e-215

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.2

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

   6. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 79.2

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

   8. Applied egg-rr 79.2%

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

   if -1.50000000000000013e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 78.6

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

   6. Applied egg-rr 78.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 78.6

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := \left(1 - \frac{h \cdot \frac{t\_0}{d}}{\ell} \cdot \frac{t\_0}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 8.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (* 0.5 M)))
        (t_1
         (*
          (- 1.0 (* (/ (* h (/ t_0 d)) l) (/ t_0 (* d 2.0))))
          (* (sqrt (/ d h)) (sqrt (/ d l))))))
   (if (<= d -1.5e-215)
     t_1
     (if (<= d -5e-310)
       (fma
        (/ 1.0 (sqrt (/ (* l l) (/ h l))))
        (* (/ (* M M) d) (* 0.125 (* D D)))
        (* (- d) (sqrt (/ 1.0 (* h l)))))
       (if (<= d 8.1e-44)
         (*
          (/ d (sqrt (* h l)))
          (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d))))))
         t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (0.5 * M);
	double t_1 = (1.0 - (((h * (t_0 / d)) / l) * (t_0 / (d * 2.0)))) * (sqrt((d / h)) * sqrt((d / l)));
	double tmp;
	if (d <= -1.5e-215) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else if (d <= 8.1e-44) {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(0.5 * M))
	t_1 = Float64(Float64(1.0 - Float64(Float64(Float64(h * Float64(t_0 / d)) / l) * Float64(t_0 / Float64(d * 2.0)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))))
	tmp = 0.0
	if (d <= -1.5e-215)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= 8.1e-44)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[(h * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.5e-215], t$95$1, If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.1e-44], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.50000000000000013e-215 or 8.0999999999999997e-44 < d

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 80.4

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

   6. Applied egg-rr 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 80.4

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

   8. Applied egg-rr 80.4%

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

   if -1.50000000000000013e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d < 8.0999999999999997e-44

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 57.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 57.6

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

   6. Applied egg-rr 57.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 57.6

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

   8. Applied egg-rr 57.6%

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

   10. Applied egg-rr 64.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 8.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d}}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3 \cdot 10^{+178}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h \cdot 0.5}{d \cdot \ell} \cdot \left(D \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= d -3e+178)
     (*
      t_0
      (*
       (sqrt (/ d l))
       (- 1.0 (* (/ (* h 0.5) (* d l)) (* D (/ (* M (* 0.25 (* D M))) d))))))
     (if (<= d -1.7e-215)
       (*
        t_0
        (*
         (/ (sqrt (- d)) (sqrt (- l)))
         (- 1.0 (/ (* (* h 0.5) (* M (* 0.25 (* D (* D M))))) (* d (* d l))))))
       (if (<= d -5e-310)
         (fma
          (/ 1.0 (sqrt (/ (* l l) (/ h l))))
          (* (/ (* M M) d) (* 0.125 (* D D)))
          (* (- d) (sqrt (/ 1.0 (* h l)))))
         (*
          (/ d (sqrt (* h l)))
          (-
           1.0
           (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (d <= -3e+178) {
		tmp = t_0 * (sqrt((d / l)) * (1.0 - (((h * 0.5) / (d * l)) * (D * ((M * (0.25 * (D * M))) / d)))));
	} else if (d <= -1.7e-215) {
		tmp = t_0 * ((sqrt(-d) / sqrt(-l)) * (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))));
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -3e+178)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(Float64(h * 0.5) / Float64(d * l)) * Float64(D * Float64(Float64(M * Float64(0.25 * Float64(D * M))) / d))))));
	elseif (d <= -1.7e-215)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 - Float64(Float64(Float64(h * 0.5) * Float64(M * Float64(0.25 * Float64(D * Float64(D * M))))) / Float64(d * Float64(d * l))))));
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3e+178], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(h * 0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(N[(M * N[(0.25 * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.7e-215], N[(t$95$0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * 0.5), $MachinePrecision] * N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.00000000000000016e178

   1. Initial program 87.3%

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

   4. Applied egg-rr 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   8. Applied egg-rr 83.4%

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

   if -3.00000000000000016e178 < d < -1.70000000000000001e-215

   1. Initial program 70.3%

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

   4. Applied egg-rr 68.1%

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

   6. Applied egg-rr 64.4%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 75.3

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

   8. Applied egg-rr 75.3%

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

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h \cdot 0.5}{d \cdot \ell} \cdot \left(D \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{-\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -9.2e-95)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* (* h (* 0.5 (* D M))) (/ (* 0.25 (* D M)) (* d (* d l)))))))
   (if (<= d -5e-310)
     (fma
      (sqrt (/ h (* l (* l l))))
      (* (/ (* M M) d) (* 0.125 (* D D)))
      (* d (/ (sqrt (/ -1.0 l)) (- (sqrt (- h))))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9.2e-95) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h * (0.5 * (D * M))) * ((0.25 * (D * M)) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = fma(sqrt((h / (l * (l * l)))), (((M * M) / d) * (0.125 * (D * D))), (d * (sqrt((-1.0 / l)) / -sqrt(-h))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -9.2e-95)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * Float64(D * M))) * Float64(Float64(0.25 * Float64(D * M)) / Float64(d * Float64(d * l)))))));
	elseif (d <= -5e-310)
		tmp = fma(sqrt(Float64(h / Float64(l * Float64(l * l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(d * Float64(sqrt(Float64(-1.0 / l)) / Float64(-sqrt(Float64(-h))))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -9.2e-95], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[(-h)], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.19999999999999997e-95

   1. Initial program 78.5%

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

   4. Applied egg-rr 71.5%

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

   6. Applied egg-rr 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied egg-rr 75.3%

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

   if -9.19999999999999997e-95 < d < -4.999999999999985e-310

   1. Initial program 39.0%

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

   4. Applied egg-rr 39.3%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 58.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{\ell}\right)}{\mathsf{neg}\left(h\right)}}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{\ell}}}{\mathsf{neg}\left(h\right)}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{\color{blue}{\frac{-1}{\ell}}}{\mathsf{neg}\left(h\right)}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{\color{blue}{\frac{-1}{\ell}}}{\mathsf{neg}\left(h\right)}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{\frac{-1}{\ell}}{\color{blue}{\mathsf{neg}\left(h\right)}}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \color{blue}{\frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \frac{\color{blue}{\sqrt{\frac{-1}{\ell}}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \frac{\sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      10. lift-/.f64 63.0

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

   9. Applied egg-rr 63.0%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{-\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2} \cdot \left(D \cdot \left(\frac{h}{\ell} \cdot \frac{0.5 \cdot M}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.5e-215)
   (*
    (- 1.0 (* (/ (* D (* 0.5 M)) (* d 2.0)) (* D (* (/ h l) (/ (* 0.5 M) d)))))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (if (<= d -5e-310)
     (fma
      (/ 1.0 (sqrt (/ (* l l) (/ h l))))
      (* (/ (* M M) d) (* 0.125 (* D D)))
      (* (- d) (sqrt (/ 1.0 (* h l)))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.5e-215) {
		tmp = (1.0 - (((D * (0.5 * M)) / (d * 2.0)) * (D * ((h / l) * ((0.5 * M) / d))))) * (sqrt((d / h)) * sqrt((d / l)));
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.5e-215)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(D * Float64(0.5 * M)) / Float64(d * 2.0)) * Float64(D * Float64(Float64(h / l) * Float64(Float64(0.5 * M) / d))))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.5e-215], N[(N[(1.0 - N[(N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(D * N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.50000000000000013e-215

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.2

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

   6. Applied egg-rr 79.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 79.2

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

   8. Applied egg-rr 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 74.0

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

   10. Applied egg-rr 74.0%

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

   if -1.50000000000000013e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2} \cdot \left(D \cdot \left(\frac{h}{\ell} \cdot \frac{0.5 \cdot M}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.7e-215)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* (* h (* 0.5 (* D M))) (/ (* 0.25 (* D M)) (* d (* d l)))))))
   (if (<= d -5e-310)
     (fma
      (/ 1.0 (sqrt (/ (* l l) (/ h l))))
      (* (/ (* M M) d) (* 0.125 (* D D)))
      (* (- d) (sqrt (/ 1.0 (* h l)))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.7e-215) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h * (0.5 * (D * M))) * ((0.25 * (D * M)) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = fma((1.0 / sqrt(((l * l) / (h / l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.7e-215)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * Float64(D * M))) * Float64(Float64(0.25 * Float64(D * M)) / Float64(d * Float64(d * l)))))));
	elseif (d <= -5e-310)
		tmp = fma(Float64(1.0 / sqrt(Float64(Float64(l * l) / Float64(h / l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.7e-215], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 / N[Sqrt[N[(N[(l * l), $MachinePrecision] / N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.70000000000000001e-215

   1. Initial program 74.1%

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

   4. Applied egg-rr 68.6%

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

   6. Applied egg-rr 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied egg-rr 70.7%

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

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}}, \frac{M \cdot M}{d} \cdot \left(\frac{1}{8} \cdot \left(D \cdot D\right)\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right) \]

      9. lower-/.f64 82.7

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

   9. Applied egg-rr 82.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\frac{\ell \cdot \ell}{\frac{h}{\ell}}}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.7e-215)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* (* h (* 0.5 (* D M))) (/ (* 0.25 (* D M)) (* d (* d l)))))))
   (if (<= d -5e-310)
     (fma
      (sqrt (/ h (* l (* l l))))
      (* (/ (* M M) d) (* 0.125 (* D D)))
      (* (- d) (sqrt (/ 1.0 (* h l)))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.7e-215) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h * (0.5 * (D * M))) * ((0.25 * (D * M)) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = fma(sqrt((h / (l * (l * l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.7e-215)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * Float64(D * M))) * Float64(Float64(0.25 * Float64(D * M)) / Float64(d * Float64(d * l)))))));
	elseif (d <= -5e-310)
		tmp = fma(sqrt(Float64(h / Float64(l * Float64(l * l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.7e-215], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.70000000000000001e-215

   1. Initial program 74.1%

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

   4. Applied egg-rr 68.6%

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

   6. Applied egg-rr 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied egg-rr 70.7%

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

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 78.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \left(0.5 \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5.5e-97)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* h (/ (* 0.125 (* D (* M (* D M)))) (* d (* d l)))))))
   (if (<= d -5e-310)
     (fma
      (sqrt (/ h (* l (* l l))))
      (* (/ (* M M) d) (* 0.125 (* D D)))
      (* (- d) (sqrt (/ 1.0 (* h l)))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.5e-97) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * ((0.125 * (D * (M * (D * M)))) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = fma(sqrt((h / (l * (l * l)))), (((M * M) / d) * (0.125 * (D * D))), (-d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5.5e-97)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(Float64(0.125 * Float64(D * Float64(M * Float64(D * M)))) / Float64(d * Float64(d * l)))))));
	elseif (d <= -5e-310)
		tmp = fma(sqrt(Float64(h / Float64(l * Float64(l * l)))), Float64(Float64(Float64(M * M) / d) * Float64(0.125 * Float64(D * D))), Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5.5e-97], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.125 * N[(D * N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.49999999999999948e-97

   1. Initial program 77.9%

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

   4. Applied egg-rr 70.9%

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

   6. Applied egg-rr 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied egg-rr 68.7%

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

   if -5.49999999999999948e-97 < d < -4.999999999999985e-310

   1. Initial program 39.4%

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

   4. Applied egg-rr 39.6%

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

   5. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Simplified 59.8%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}, \frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - M \cdot \left(\frac{D \cdot M}{d \cdot \left(d \cdot \ell\right)} \cdot \left(0.125 \cdot \left(h \cdot D\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.3e+145)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (if (<= d -1.7e-215)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* M (* (/ (* D M) (* d (* d l))) (* 0.125 (* h D)))))))
     (if (<= d -5e-310)
       (* (* M M) (* (sqrt (/ h (* l (* l l)))) (* (* D D) (/ 0.125 d))))
       (*
        (/ d (sqrt (* h l)))
        (-
         1.0
         (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.3e+145) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -1.7e-215) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (M * (((D * M) / (d * (d * l))) * (0.125 * (h * D))))));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-2.3d+145)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (d <= (-1.7d-215)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (m * (((d_1 * m) / (d * (d * l))) * (0.125d0 * (h * d_1))))))
    else if (d <= (-5d-310)) then
        tmp = (m * m) * (sqrt((h / (l * (l * l)))) * ((d_1 * d_1) * (0.125d0 / d)))
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 - (((d_1 * (h * (0.5d0 * m))) / (d * l)) * (d_1 * (0.25d0 * (m / d)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.3e+145) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (d <= -1.7e-215) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (M * (((D * M) / (d * (d * l))) * (0.125 * (h * D))))));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (Math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.3e+145:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif d <= -1.7e-215:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (M * (((D * M) / (d * (d * l))) * (0.125 * (h * D))))))
	elif d <= -5e-310:
		tmp = (M * M) * (math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)))
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.3e+145)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -1.7e-215)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(M * Float64(Float64(Float64(D * M) / Float64(d * Float64(d * l))) * Float64(0.125 * Float64(h * D)))))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(M * M) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * D) * Float64(0.125 / d))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.3e+145)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (d <= -1.7e-215)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (M * (((D * M) / (d * (d * l))) * (0.125 * (h * D))))));
	elseif (d <= -5e-310)
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	else
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.3e+145], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.7e-215], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(M * N[(N[(N[(D * M), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(M * M), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.3e145

   1. Initial program 79.0%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 71.7

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

   7. Simplified 71.7%

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

   if -2.3e145 < d < -1.70000000000000001e-215

   1. Initial program 72.3%

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

   4. Applied egg-rr 69.9%

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

   6. Applied egg-rr 67.2%

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

   8. Applied egg-rr 69.5%

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

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 22.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 22.6

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

   6. Applied egg-rr 22.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 22.6

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

   8. Applied egg-rr 22.6%

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

   9. Taylor expanded in h around -inf

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

   11. Simplified 62.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - M \cdot \left(\frac{D \cdot M}{d \cdot \left(d \cdot \ell\right)} \cdot \left(0.125 \cdot \left(h \cdot D\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2200:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2200.0)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -1.7e-215)
     (*
      (- 1.0 (/ (* (* h 0.5) (* M (* 0.25 (* D (* D M))))) (* d (* d l))))
      (sqrt (/ (* d d) (* h l))))
     (if (<= d -5e-310)
       (* (* M M) (* (sqrt (/ h (* l (* l l)))) (* (* D D) (/ 0.125 d))))
       (*
        (/ d (sqrt (* h l)))
        (-
         1.0
         (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2200.0) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -1.7e-215) {
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-2200.0d0)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-1.7d-215)) then
        tmp = (1.0d0 - (((h * 0.5d0) * (m * (0.25d0 * (d_1 * (d_1 * m))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)))
    else if (d <= (-5d-310)) then
        tmp = (m * m) * (sqrt((h / (l * (l * l)))) * ((d_1 * d_1) * (0.125d0 / d)))
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 - (((d_1 * (h * (0.5d0 * m))) / (d * l)) * (d_1 * (0.25d0 * (m / d)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2200.0) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -1.7e-215) {
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * Math.sqrt(((d * d) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (Math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2200.0:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -1.7e-215:
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * math.sqrt(((d * d) / (h * l)))
	elif d <= -5e-310:
		tmp = (M * M) * (math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)))
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2200.0)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -1.7e-215)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h * 0.5) * Float64(M * Float64(0.25 * Float64(D * Float64(D * M))))) / Float64(d * Float64(d * l)))) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(M * M) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * D) * Float64(0.125 / d))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2200.0)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -1.7e-215)
		tmp = (1.0 - (((h * 0.5) * (M * (0.25 * (D * (D * M))))) / (d * (d * l)))) * sqrt(((d * d) / (h * l)));
	elseif (d <= -5e-310)
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	else
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2200.0], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.7e-215], N[(N[(1.0 - N[(N[(N[(h * 0.5), $MachinePrecision] * N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(M * M), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2200

   1. Initial program 80.6%

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

   4. Applied egg-rr 70.1%

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

   5. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 2.0

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

   7. Simplified 2.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 2.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 2.0

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

   9. Applied egg-rr 2.0%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. pow1/2 N/A

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

       4. frac-times N/A

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

       5. sqrt-div N/A

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

       6. lift-/.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. pow1/2 N/A

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

       9. sqrt-div N/A

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

       10. lift-/.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 66.3

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

   11. Applied egg-rr 66.3%

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

   if -2200 < d < -1.70000000000000001e-215

   1. Initial program 66.5%

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

   4. Applied egg-rr 66.8%

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

   6. Applied egg-rr 46.3%

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

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 22.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 22.6

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

   6. Applied egg-rr 22.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 22.6

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

   8. Applied egg-rr 22.6%

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

   9. Taylor expanded in h around -inf

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

   11. Simplified 62.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*r* N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 70.8

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

   6. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 70.8

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

   8. Applied egg-rr 70.8%

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

   10. Applied egg-rr 68.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2200:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\left(1 - \frac{\left(h \cdot 0.5\right) \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.7e-215)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* h (/ (* 0.125 (* D (* M (* D M)))) (* d (* d l)))))))
   (if (<= d -5e-310)
     (* (* M M) (* (sqrt (/ h (* l (* l l)))) (* (* D D) (/ 0.125 d))))
     (*
      (/ d (sqrt (* h l)))
      (- 1.0 (* (/ (* D (* h (* 0.5 M))) (* d l)) (* D (* 0.25 (/ M d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.7e-215) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * ((0.125 * (D * (M * (D * M)))) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-1.7d-215)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (h * ((0.125d0 * (d_1 * (m * (d_1 * m)))) / (d * (d * l))))))
    else if (d <= (-5d-310)) then
        tmp = (m * m) * (sqrt((h / (l * (l * l)))) * ((d_1 * d_1) * (0.125d0 / d)))
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 - (((d_1 * (h * (0.5d0 * m))) / (d * l)) * (d_1 * (0.25d0 * (m / d)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.7e-215) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * ((0.125 * (D * (M * (D * M)))) / (d * (d * l))))));
	} else if (d <= -5e-310) {
		tmp = (M * M) * (Math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.7e-215:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * ((0.125 * (D * (M * (D * M)))) / (d * (d * l))))))
	elif d <= -5e-310:
		tmp = (M * M) * (math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)))
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.7e-215)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(Float64(0.125 * Float64(D * Float64(M * Float64(D * M)))) / Float64(d * Float64(d * l)))))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(M * M) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * D) * Float64(0.125 / d))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(0.5 * M))) / Float64(d * l)) * Float64(D * Float64(0.25 * Float64(M / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.7e-215)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * ((0.125 * (D * (M * (D * M)))) / (d * (d * l))))));
	elseif (d <= -5e-310)
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	else
		tmp = (d / sqrt((h * l))) * (1.0 - (((D * (h * (0.5 * M))) / (d * l)) * (D * (0.25 * (M / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.7e-215], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.125 * N[(D * N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(M * M), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D * N[(h * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D * N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.70000000000000001e-215

   1. Initial program 74.1%

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

   4. Applied egg-rr 68.6%

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

   6. Applied egg-rr 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \frac{1}{4}\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      10. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{h \cdot \frac{\frac{1}{2} \cdot \left(M \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \frac{1}{4}\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \frac{1}{4}\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \frac{1}{4}\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   8. Applied egg-rr 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   if -1.70000000000000001e-215 < d < -4.999999999999985e-310

   1. Initial program 22.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

   4. Applied egg-rr 22.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lower-sqrt.f64 22.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   6. Applied egg-rr 22.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lift-sqrt.f64 22.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   8. Applied egg-rr 22.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   11. Simplified 62.3%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

   4. Applied egg-rr 70.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lower-sqrt.f64 70.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   6. Applied egg-rr 70.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   8. Applied egg-rr 70.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{D \cdot \left(\left(0.5 \cdot M\right) \cdot h\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{D \cdot \left(h \cdot \left(0.5 \cdot M\right)\right)}{d \cdot \ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-292}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot t\_0\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* l (* l l))))))
   (if (<= d -1.95e+175)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d -7.2e-110)
       (/ (sqrt (* d (/ d (- h)))) (sqrt (- l)))
       (if (<= d -7.8e-292)
         (/ (* t_0 (* (* M M) (* 0.125 (* D D)))) d)
         (if (<= d 3.2e-192)
           (/ (* (* (* M M) -0.125) (* (* D D) t_0)) d)
           (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -7.2e-110) {
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	} else if (d <= -7.8e-292) {
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	} else if (d <= 3.2e-192) {
		tmp = (((M * M) * -0.125) * ((D * D) * t_0)) / d;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / (l * (l * l))))
    if (d <= (-1.95d+175)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-7.2d-110)) then
        tmp = sqrt((d * (d / -h))) / sqrt(-l)
    else if (d <= (-7.8d-292)) then
        tmp = (t_0 * ((m * m) * (0.125d0 * (d_1 * d_1)))) / d
    else if (d <= 3.2d-192) then
        tmp = (((m * m) * (-0.125d0)) * ((d_1 * d_1) * t_0)) / d
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -7.2e-110) {
		tmp = Math.sqrt((d * (d / -h))) / Math.sqrt(-l);
	} else if (d <= -7.8e-292) {
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	} else if (d <= 3.2e-192) {
		tmp = (((M * M) * -0.125) * ((D * D) * t_0)) / d;
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if d <= -1.95e+175:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -7.2e-110:
		tmp = math.sqrt((d * (d / -h))) / math.sqrt(-l)
	elif d <= -7.8e-292:
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d
	elif d <= 3.2e-192:
		tmp = (((M * M) * -0.125) * ((D * D) * t_0)) / d
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (d <= -1.95e+175)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -7.2e-110)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -7.8e-292)
		tmp = Float64(Float64(t_0 * Float64(Float64(M * M) * Float64(0.125 * Float64(D * D)))) / d);
	elseif (d <= 3.2e-192)
		tmp = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D * D) * t_0)) / d);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (d <= -1.95e+175)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -7.2e-110)
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	elseif (d <= -7.8e-292)
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	elseif (d <= 3.2e-192)
		tmp = (((M * M) * -0.125) * ((D * D) * t_0)) / d;
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e+175], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.2e-110], N[(N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.8e-292], N[(N[(t$95$0 * N[(N[(M * M), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.2e-192], N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.94999999999999986e175

   1. Initial program 87.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 70.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 1.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 1.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 1.7

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 1.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       9. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       11. lift-sqrt.f64 N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       13. lower-*.f64 74.8

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   11. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.94999999999999986e175 < d < -7.1999999999999999e-110

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 71.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 4.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 4.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 2.9

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 2.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       13. lift-neg.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

       14. associate-*r/ N/A

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       15. sqrt-div N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       17. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       19. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       20. lower-neg.f64 54.5

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\color{blue}{-\ell}}} \]

   11. Applied egg-rr 54.5%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]

   if -7.1999999999999999e-110 < d < -7.8e-292

   1. Initial program 36.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 37.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   7. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}} \]

   if -7.8e-292 < d < 3.2000000000000002e-192

   1. Initial program 42.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

   4. Applied egg-rr 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lower-sqrt.f64 45.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   6. Applied egg-rr 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lift-sqrt.f64 45.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   8. Applied egg-rr 45.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   9. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \]

       3. associate-*r* N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot {M}^{2}}}{d} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}}{d} \cdot {M}^{2}\right)} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2}}{d}\right)} \cdot {M}^{2}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \cdot {M}^{2}\right) \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot {M}^{2}} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

       9. associate-*r* N/A

          \[\leadsto \color{blue}{\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

       10. *-commutative N/A

           \[\leadsto \left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2}}{d}\right)} \]

       11. associate-*r/ N/A

           \[\leadsto \left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}}{d}} \]

       12. associate-*r/ N/A

           \[\leadsto \color{blue}{\frac{\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right)}{d}} \]

       13. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right)}{d}} \]

   11. Simplified 45.8%

       \[\leadsto \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}} \]

   if 3.2000000000000002e-192 < d

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 68.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 49.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 49.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 60.7

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-292}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* l (* l l))))))
   (if (<= d -1.95e+175)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d -7.2e-110)
       (/ (sqrt (* d (/ d (- h)))) (sqrt (- l)))
       (if (<= d -7.8e-292)
         (/ (* t_0 (* (* M M) (* 0.125 (* D D)))) d)
         (if (<= d 3.2e-192)
           (/ (* t_0 (* (* D D) (* (* M M) -0.125))) d)
           (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -7.2e-110) {
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	} else if (d <= -7.8e-292) {
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	} else if (d <= 3.2e-192) {
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / (l * (l * l))))
    if (d <= (-1.95d+175)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-7.2d-110)) then
        tmp = sqrt((d * (d / -h))) / sqrt(-l)
    else if (d <= (-7.8d-292)) then
        tmp = (t_0 * ((m * m) * (0.125d0 * (d_1 * d_1)))) / d
    else if (d <= 3.2d-192) then
        tmp = (t_0 * ((d_1 * d_1) * ((m * m) * (-0.125d0)))) / d
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -7.2e-110) {
		tmp = Math.sqrt((d * (d / -h))) / Math.sqrt(-l);
	} else if (d <= -7.8e-292) {
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	} else if (d <= 3.2e-192) {
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if d <= -1.95e+175:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -7.2e-110:
		tmp = math.sqrt((d * (d / -h))) / math.sqrt(-l)
	elif d <= -7.8e-292:
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d
	elif d <= 3.2e-192:
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (d <= -1.95e+175)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -7.2e-110)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -7.8e-292)
		tmp = Float64(Float64(t_0 * Float64(Float64(M * M) * Float64(0.125 * Float64(D * D)))) / d);
	elseif (d <= 3.2e-192)
		tmp = Float64(Float64(t_0 * Float64(Float64(D * D) * Float64(Float64(M * M) * -0.125))) / d);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (d <= -1.95e+175)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -7.2e-110)
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	elseif (d <= -7.8e-292)
		tmp = (t_0 * ((M * M) * (0.125 * (D * D)))) / d;
	elseif (d <= 3.2e-192)
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e+175], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.2e-110], N[(N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.8e-292], N[(N[(t$95$0 * N[(N[(M * M), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.2e-192], N[(N[(t$95$0 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.94999999999999986e175

   1. Initial program 87.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 70.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 1.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 1.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 1.7

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 1.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       9. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       11. lift-sqrt.f64 N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       13. lower-*.f64 74.8

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   11. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.94999999999999986e175 < d < -7.1999999999999999e-110

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 71.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 4.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 4.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 2.9

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 2.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       13. lift-neg.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

       14. associate-*r/ N/A

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       15. sqrt-div N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       17. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       19. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       20. lower-neg.f64 54.5

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\color{blue}{-\ell}}} \]

   11. Applied egg-rr 54.5%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]

   if -7.1999999999999999e-110 < d < -7.8e-292

   1. Initial program 36.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 37.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   7. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}} \]

   if -7.8e-292 < d < 3.2000000000000002e-192

   1. Initial program 42.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 42.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}} \]

   if 3.2000000000000002e-192 < d

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 68.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 49.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 49.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 60.7

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-308}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(t\_0 \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* l (* l l))))))
   (if (<= d -1.95e+175)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d -8e-112)
       (/ (sqrt (* d (/ d (- h)))) (sqrt (- l)))
       (if (<= d 6.5e-308)
         (* (* M M) (* t_0 (* (* D D) (/ 0.125 d))))
         (if (<= d 3.2e-192)
           (/ (* t_0 (* (* D D) (* (* M M) -0.125))) d)
           (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -8e-112) {
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	} else if (d <= 6.5e-308) {
		tmp = (M * M) * (t_0 * ((D * D) * (0.125 / d)));
	} else if (d <= 3.2e-192) {
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / (l * (l * l))))
    if (d <= (-1.95d+175)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-8d-112)) then
        tmp = sqrt((d * (d / -h))) / sqrt(-l)
    else if (d <= 6.5d-308) then
        tmp = (m * m) * (t_0 * ((d_1 * d_1) * (0.125d0 / d)))
    else if (d <= 3.2d-192) then
        tmp = (t_0 * ((d_1 * d_1) * ((m * m) * (-0.125d0)))) / d
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (d <= -1.95e+175) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -8e-112) {
		tmp = Math.sqrt((d * (d / -h))) / Math.sqrt(-l);
	} else if (d <= 6.5e-308) {
		tmp = (M * M) * (t_0 * ((D * D) * (0.125 / d)));
	} else if (d <= 3.2e-192) {
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if d <= -1.95e+175:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -8e-112:
		tmp = math.sqrt((d * (d / -h))) / math.sqrt(-l)
	elif d <= 6.5e-308:
		tmp = (M * M) * (t_0 * ((D * D) * (0.125 / d)))
	elif d <= 3.2e-192:
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (d <= -1.95e+175)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -8e-112)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= 6.5e-308)
		tmp = Float64(Float64(M * M) * Float64(t_0 * Float64(Float64(D * D) * Float64(0.125 / d))));
	elseif (d <= 3.2e-192)
		tmp = Float64(Float64(t_0 * Float64(Float64(D * D) * Float64(Float64(M * M) * -0.125))) / d);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (d <= -1.95e+175)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -8e-112)
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	elseif (d <= 6.5e-308)
		tmp = (M * M) * (t_0 * ((D * D) * (0.125 / d)));
	elseif (d <= 3.2e-192)
		tmp = (t_0 * ((D * D) * ((M * M) * -0.125))) / d;
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e+175], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-112], N[(N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e-308], N[(N[(M * M), $MachinePrecision] * N[(t$95$0 * N[(N[(D * D), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-192], N[(N[(t$95$0 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.94999999999999986e175

   1. Initial program 87.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 70.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 1.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 1.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 1.7

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 1.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       9. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       11. lift-sqrt.f64 N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       13. lower-*.f64 74.8

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   11. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.94999999999999986e175 < d < -7.9999999999999996e-112

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 71.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 4.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 4.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 2.9

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 2.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       13. lift-neg.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

       14. associate-*r/ N/A

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       15. sqrt-div N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       17. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       19. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       20. lower-neg.f64 54.5

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\color{blue}{-\ell}}} \]

   11. Applied egg-rr 54.5%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]

   if -7.9999999999999996e-112 < d < 6.4999999999999999e-308

   1. Initial program 36.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

   4. Applied egg-rr 39.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lower-sqrt.f64 39.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   6. Applied egg-rr 39.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lift-sqrt.f64 39.2

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   8. Applied egg-rr 39.2%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   11. Simplified 51.5%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)} \]

   if 6.4999999999999999e-308 < d < 3.2000000000000002e-192

   1. Initial program 43.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 44.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}} \]

   if 3.2000000000000002e-192 < d

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 68.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 49.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 49.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 60.7

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-308}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.95e+175)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -8e-112)
     (/ (sqrt (* d (/ d (- h)))) (sqrt (- l)))
     (if (<= d -5e-310)
       (* (* M M) (* (sqrt (/ h (* l (* l l)))) (* (* D D) (/ 0.125 d))))
       (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.95e+175) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -8e-112) {
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	} else if (d <= -5e-310) {
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-1.95d+175)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-8d-112)) then
        tmp = sqrt((d * (d / -h))) / sqrt(-l)
    else if (d <= (-5d-310)) then
        tmp = (m * m) * (sqrt((h / (l * (l * l)))) * ((d_1 * d_1) * (0.125d0 / d)))
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.95e+175) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -8e-112) {
		tmp = Math.sqrt((d * (d / -h))) / Math.sqrt(-l);
	} else if (d <= -5e-310) {
		tmp = (M * M) * (Math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.95e+175:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -8e-112:
		tmp = math.sqrt((d * (d / -h))) / math.sqrt(-l)
	elif d <= -5e-310:
		tmp = (M * M) * (math.sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)))
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.95e+175)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -8e-112)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(M * M) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * D) * Float64(0.125 / d))));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.95e+175)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -8e-112)
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	elseif (d <= -5e-310)
		tmp = (M * M) * (sqrt((h / (l * (l * l)))) * ((D * D) * (0.125 / d)));
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.95e+175], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-112], N[(N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(M * M), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.94999999999999986e175

   1. Initial program 87.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 70.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 1.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 1.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 1.7

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 1.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       9. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       11. lift-sqrt.f64 N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       13. lower-*.f64 74.8

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   11. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.94999999999999986e175 < d < -7.9999999999999996e-112

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 71.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 4.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 4.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 2.9

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 2.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       13. lift-neg.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

       14. associate-*r/ N/A

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       15. sqrt-div N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       17. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       19. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       20. lower-neg.f64 54.5

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\color{blue}{-\ell}}} \]

   11. Applied egg-rr 54.5%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]

   if -7.9999999999999996e-112 < d < -4.999999999999985e-310

   1. Initial program 37.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

   4. Applied egg-rr 40.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lower-sqrt.f64 40.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   6. Applied egg-rr 40.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(\frac{1}{2} \cdot M\right)}{d \cdot 2}\right) \]

      4. lift-sqrt.f64 40.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   8. Applied egg-rr 40.0%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot 2}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   11. Simplified 52.6%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)} \]

   if -4.999999999999985e-310 < d

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 40.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 40.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 50.7

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 50.7%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.45e+228)
   (/ (sqrt (* d (/ d (- h)))) (sqrt (- l)))
   (if (<= l 2.5e-241)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (* d (/ (/ 1.0 (sqrt h)) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.45e+228) {
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	} else if (l <= 2.5e-241) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-1.45d+228)) then
        tmp = sqrt((d * (d / -h))) / sqrt(-l)
    else if (l <= 2.5d-241) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.45e+228) {
		tmp = Math.sqrt((d * (d / -h))) / Math.sqrt(-l);
	} else if (l <= 2.5e-241) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.45e+228:
		tmp = math.sqrt((d * (d / -h))) / math.sqrt(-l)
	elif l <= 2.5e-241:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.45e+228)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(-h)))) / sqrt(Float64(-l)));
	elseif (l <= 2.5e-241)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.45e+228)
		tmp = sqrt((d * (d / -h))) / sqrt(-l);
	elseif (l <= 2.5e-241)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.45e+228], N[(N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-241], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.45000000000000001e228

   1. Initial program 54.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 49.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 26.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 26.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 26.6

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 26.6

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       13. lift-neg.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

       14. associate-*r/ N/A

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       15. sqrt-div N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       17. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

       19. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

       20. lower-neg.f64 53.9

           \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\color{blue}{-\ell}}} \]

   11. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]

   if -1.45000000000000001e228 < l < 2.4999999999999999e-241

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      11. lower-neg.f64 48.0

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 2.4999999999999999e-241 < l

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 42.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 42.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 53.1

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 53.1%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{d \cdot \frac{d}{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.5e-241)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-241) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 2.5d-241) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-241) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.5e-241:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.5e-241)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.5e-241)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.5e-241], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4999999999999999e-241

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 61.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      11. lower-neg.f64 45.2

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 2.4999999999999999e-241 < l

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 42.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 42.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/l/ N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      5. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{h}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

      7. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      8. sqrt-div N/A

         \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{h}}}}{\sqrt{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{h}}}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

      12. lower-sqrt.f64 53.1

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   9. Applied egg-rr 53.1%

      \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 44.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.5e-241)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-241) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 2.5d-241) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-241) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.5e-241:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.5e-241)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.5e-241)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.5e-241], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4999999999999999e-241

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 61.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      11. lower-neg.f64 45.2

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 2.4999999999999999e-241 < l

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 42.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 42.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 43.4

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 43.4

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 43.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{{h}^{\frac{1}{2}}}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot {h}^{\frac{1}{2}}}} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\frac{1}{2}}} \]

       6. pow1/2 N/A

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       7. lower-sqrt.f64 53.1

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   11. Applied egg-rr 53.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 41.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 5e-241) (* (- d) (sqrt (/ 1.0 (* h l)))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5e-241) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 5d-241) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5e-241) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 5e-241:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 5e-241)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 5e-241)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 5e-241], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999998e-241

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 61.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      11. lower-neg.f64 45.2

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 4.9999999999999998e-241 < l

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 62.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 42.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 42.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 43.4

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 43.4

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 43.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 33.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.15e-112) (sqrt (/ (* d d) (* h l))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.15e-112) {
		tmp = sqrt(((d * d) / (h * l)));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-1.15d-112)) then
        tmp = sqrt(((d * d) / (h * l)))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.15e-112) {
		tmp = Math.sqrt(((d * d) / (h * l)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.15e-112:
		tmp = math.sqrt(((d * d) / (h * l)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.15e-112)
		tmp = sqrt(Float64(Float64(d * d) / Float64(h * l)));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.15e-112)
		tmp = sqrt(((d * d) / (h * l)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.15e-112], N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.14999999999999995e-112

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 71.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 3.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 3.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 2.5

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 2.5

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}} \cdot \sqrt{\ell}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       8. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       10. sqrt-unprod N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       12. lift-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

       14. frac-times N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

       15. *-commutative N/A

           \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \]

       17. lower-/.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \]

       18. lower-*.f64 31.9

           \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \]

   11. Applied egg-rr 31.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{\ell \cdot h}}} \]

   if -1.14999999999999995e-112 < d

   1. Initial program 59.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 56.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 34.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 34.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 34.6

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 34.6

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 25.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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 / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

4. Applied egg-rr 61.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.25}{d} \cdot \frac{0.5 \cdot h}{\ell}}{d}}\right) \]

5. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. *-commutative N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. lower-*.f64 24.3

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

7. Simplified 24.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

   3. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   6. lower-sqrt.f64 24.1

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. lower-*.f64 24.1

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

9. Applied egg-rr 24.1%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))