Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 80.3%

Time: 21.5s

Alternatives: 29

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d \cdot 2}}{\ell}\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - t\_1 \cdot \left(h \cdot \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_0, d \cdot t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 + t\_1 \cdot \frac{\frac{M\_m \cdot D\_m}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))) (t_1 (/ (/ (* M_m (* D_m 0.5)) (* d 2.0)) l)))
   (if (<= d -8.2e-193)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
      (- 1.0 (* t_1 (* h (/ (* 0.5 (* M_m D_m)) d)))))
     (if (<= d 8e-289)
       (/
        (fma
         (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
         t_0
         (* d t_0))
        h)
       (*
        (* (pow (/ d h) (/ 1.0 2.0)) (* (sqrt (/ 1.0 l)) (sqrt d)))
        (+ 1.0 (* t_1 (/ (/ (* M_m D_m) (* d 2.0)) (/ -1.0 h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = ((M_m * (D_m * 0.5)) / (d * 2.0)) / l;
	double tmp;
	if (d <= -8.2e-193) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - (t_1 * (h * ((0.5 * (M_m * D_m)) / d))));
	} else if (d <= 8e-289) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_0, (d * t_0)) / h;
	} else {
		tmp = (pow((d / h), (1.0 / 2.0)) * (sqrt((1.0 / l)) * sqrt(d))) * (1.0 + (t_1 * (((M_m * D_m) / (d * 2.0)) / (-1.0 / h))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d * 2.0)) / l)
	tmp = 0.0
	if (d <= -8.2e-193)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(t_1 * Float64(h * Float64(Float64(0.5 * Float64(M_m * D_m)) / d)))));
	elseif (d <= 8e-289)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_0, Float64(d * t_0)) / h);
	else
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(1.0 / l)) * sqrt(d))) * Float64(1.0 + Float64(t_1 * Float64(Float64(Float64(M_m * D_m) / Float64(d * 2.0)) / Float64(-1.0 / h)))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -8.2e-193], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h * N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-289], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(d * t$95$0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.20000000000000005e-193

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 80.1

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 80.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 91.4

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

   10. Applied rewrites 91.4%

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

   if -8.20000000000000005e-193 < d < 8.0000000000000001e-289

   1. Initial program 31.1%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 59.8%

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

   if 8.0000000000000001e-289 < d

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 85.9

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

   6. Applied rewrites 85.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 + \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_2 := \frac{d \cdot t\_0}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_2 (/ (* d t_0) h)))
   (if (<= t_1 -2e-180)
     (/ (* (- d) t_0) h)
     (if (<= t_1 2e-158)
       t_2
       (if (<= t_1 2e+136)
         (sqrt (* (/ d l) (/ d h)))
         (if (<= t_1 INFINITY) t_2 (* (- d) (sqrt (/ 1.0 (* h l))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_2 = (d * t_0) / h;
	double tmp;
	if (t_1 <= -2e-180) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 2e-158) {
		tmp = t_2;
	} else if (t_1 <= 2e+136) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_2 = (d * t_0) / h;
	double tmp;
	if (t_1 <= -2e-180) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 2e-158) {
		tmp = t_2;
	} else if (t_1 <= 2e+136) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_2 = (d * t_0) / h
	tmp = 0
	if t_1 <= -2e-180:
		tmp = (-d * t_0) / h
	elif t_1 <= 2e-158:
		tmp = t_2
	elif t_1 <= 2e+136:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_2 = Float64(Float64(d * t_0) / h)
	tmp = 0.0
	if (t_1 <= -2e-180)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif (t_1 <= 2e-158)
		tmp = t_2;
	elseif (t_1 <= 2e+136)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_2 = (d * t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-180)
		tmp = (-d * t_0) / h;
	elseif (t_1 <= 2e-158)
		tmp = t_2;
	elseif (t_1 <= 2e+136)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-180], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 2e-158], t$95$2, If[LessEqual[t$95$1, 2e+136], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-180

   1. Initial program 86.4%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 22.4%

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

   if -2e-180 < (*.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)))) < 2.00000000000000013e-158 or 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 63.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 26.3%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 75.6%

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

   if 2.00000000000000013e-158 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 99.0%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 58.4%

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

   9. Applied rewrites 99.1%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{d \cdot t\_2}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, t\_2, \left(D\_m \cdot \frac{D\_m \cdot \left(M\_m \cdot M\_m\right)}{d}\right) \cdot \left(-0.125 \cdot \frac{h \cdot t\_2}{\ell}\right)\right)}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ h l)))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_3 -2e-180)
     (*
      t_0
      (*
       t_1
       (fma
        (/ (* M_m (* D_m (* M_m D_m))) (* 4.0 (* d d)))
        (* (/ h l) -0.5)
        1.0)))
     (if (<= t_3 2e+136)
       (* t_1 t_0)
       (if (<= t_3 INFINITY)
         (/ (* d t_2) h)
         (/
          (fma
           d
           t_2
           (* (* D_m (/ (* D_m (* M_m M_m)) d)) (* -0.125 (/ (* h t_2) l))))
          h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / l));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_3 <= -2e-180) {
		tmp = t_0 * (t_1 * fma(((M_m * (D_m * (M_m * D_m))) / (4.0 * (d * d))), ((h / l) * -0.5), 1.0));
	} else if (t_3 <= 2e+136) {
		tmp = t_1 * t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (d * t_2) / h;
	} else {
		tmp = fma(d, t_2, ((D_m * ((D_m * (M_m * M_m)) / d)) * (-0.125 * ((h * t_2) / l)))) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / l))
	t_3 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_3 <= -2e-180)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(Float64(M_m * Float64(D_m * Float64(M_m * D_m))) / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0)));
	elseif (t_3 <= 2e+136)
		tmp = Float64(t_1 * t_0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(d * t_2) / h);
	else
		tmp = Float64(fma(d, t_2, Float64(Float64(D_m * Float64(Float64(D_m * Float64(M_m * M_m)) / d)) * Float64(-0.125 * Float64(Float64(h * t_2) / l)))) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-180], N[(t$95$0 * N[(t$95$1 * N[(N[(N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+136], N[(t$95$1 * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(d * t$95$2), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * t$95$2 + N[(N[(D$95$m * N[(N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(N[(h * t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-180

   1. Initial program 86.4%

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

   4. Applied rewrites 66.6%

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

   if -2e-180 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 90.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 90.3%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 10.7%

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

   7. Applied rewrites 19.1%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \frac{h \cdot \sqrt{\frac{h}{\ell}}}{\ell}\right)\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_2 -2e-180)
     (*
      t_0
      (*
       t_1
       (fma
        (/ (* M_m (* D_m (* M_m D_m))) (* 4.0 (* d d)))
        (* (/ h l) -0.5)
        1.0)))
     (if (<= t_2 2e+136)
       (* t_1 t_0)
       (if (<= t_2 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_2 <= -2e-180) {
		tmp = t_0 * (t_1 * fma(((M_m * (D_m * (M_m * D_m))) / (4.0 * (d * d))), ((h / l) * -0.5), 1.0));
	} else if (t_2 <= 2e+136) {
		tmp = t_1 * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= -2e-180)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(Float64(M_m * Float64(D_m * Float64(M_m * D_m))) / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0)));
	elseif (t_2 <= 2e+136)
		tmp = Float64(t_1 * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-180], N[(t$95$0 * N[(t$95$1 * N[(N[(N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+136], N[(t$95$1 * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-180

   1. Initial program 86.4%

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

   4. Applied rewrites 66.6%

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

   if -2e-180 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 90.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 90.3%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_2 -2e-180)
     (*
      t_0
      (*
       t_1
       (fma
        (/ (* M_m (* D_m (* M_m D_m))) (* 4.0 (* d d)))
        (* (/ h l) -0.5)
        1.0)))
     (if (<= t_2 2e+136)
       (* t_0 t_1)
       (if (<= t_2 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_2 <= -2e-180) {
		tmp = t_0 * (t_1 * fma(((M_m * (D_m * (M_m * D_m))) / (4.0 * (d * d))), ((h / l) * -0.5), 1.0));
	} else if (t_2 <= 2e+136) {
		tmp = t_0 * t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= -2e-180)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(Float64(M_m * Float64(D_m * Float64(M_m * D_m))) / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0)));
	elseif (t_2 <= 2e+136)
		tmp = Float64(t_0 * t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-180], N[(t$95$0 * N[(t$95$1 * N[(N[(N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+136], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-180

   1. Initial program 86.4%

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

   4. Applied rewrites 66.6%

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

   if -2e-180 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 90.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 90.3%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -2e-99)
     (/
      (* t_0 (/ (* (* D_m D_m) (* -0.125 (* h (* M_m M_m)))) (* d (* d l))))
      (sqrt (/ l d)))
     (if (<= t_1 2e+136)
       (* (sqrt (/ d l)) t_0)
       (if (<= t_1 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -2e-99) {
		tmp = (t_0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)))) / sqrt((l / d));
	} else if (t_1 <= 2e+136) {
		tmp = sqrt((d / l)) * t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -2e-99) {
		tmp = (t_0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)))) / Math.sqrt((l / d));
	} else if (t_1 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_1 <= -2e-99:
		tmp = (t_0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)))) / math.sqrt((l / d))
	elif t_1 <= 2e+136:
		tmp = math.sqrt((d / l)) * t_0
	elif t_1 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -2e-99)
		tmp = Float64(Float64(t_0 * Float64(Float64(Float64(D_m * D_m) * Float64(-0.125 * Float64(h * Float64(M_m * M_m)))) / Float64(d * Float64(d * l)))) / sqrt(Float64(l / d)));
	elseif (t_1 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * t_0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / h));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_1 <= -2e-99)
		tmp = (t_0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)))) / sqrt((l / d));
	elseif (t_1 <= 2e+136)
		tmp = sqrt((d / l)) * t_0;
	elseif (t_1 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-99], N[(N[(t$95$0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(-0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-99

   1. Initial program 86.1%

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-99 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 90.8%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 87.7%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ h (* l l)) l)) (/ (* (* M_m (* D_m (* M_m D_m))) 0.125) d))
     (if (<= t_0 2e+136)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((h / (l * l)) / l)) * (((M_m * (D_m * (M_m * D_m))) * 0.125) / d);
	} else if (t_0 <= 2e+136) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((h / (l * l)) / l)) * (((M_m * (D_m * (M_m * D_m))) * 0.125) / d);
	} else if (t_0 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((h / (l * l)) / l)) * (((M_m * (D_m * (M_m * D_m))) * 0.125) / d)
	elif t_0 <= 2e+136:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(Float64(M_m * Float64(D_m * Float64(M_m * D_m))) * 0.125) / d));
	elseif (t_0 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((h / (l * l)) / l)) * (((M_m * (D_m * (M_m * D_m))) * 0.125) / d);
	elseif (t_0 <= 2e+136)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))) < 0.0

   1. Initial program 82.8%

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-in N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 28.5%

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

   9. Applied rewrites 30.5%

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

   11. Applied rewrites 40.3%

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

   if 0.0 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 97.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 97.1%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(0.125 \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right) \cdot \left(D\_m \cdot \frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_0 0.0)
     (* (* 0.125 (sqrt (/ h (* l (* l l))))) (* D_m (/ (* M_m (* M_m D_m)) d)))
     (if (<= t_0 2e+136)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (0.125 * sqrt((h / (l * (l * l))))) * (D_m * ((M_m * (M_m * D_m)) / d));
	} else if (t_0 <= 2e+136) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (0.125 * Math.sqrt((h / (l * (l * l))))) * (D_m * ((M_m * (M_m * D_m)) / d));
	} else if (t_0 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (0.125 * math.sqrt((h / (l * (l * l))))) * (D_m * ((M_m * (M_m * D_m)) / d))
	elif t_0 <= 2e+136:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(0.125 * sqrt(Float64(h / Float64(l * Float64(l * l))))) * Float64(D_m * Float64(Float64(M_m * Float64(M_m * D_m)) / d)));
	elseif (t_0 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (0.125 * sqrt((h / (l * (l * l))))) * (D_m * ((M_m * (M_m * D_m)) / d));
	elseif (t_0 <= 2e+136)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(0.125 * N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))) < 0.0

   1. Initial program 82.8%

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-in N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 28.5%

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

   9. Applied rewrites 36.5%

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

   if 0.0 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 97.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 97.1%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 0:\\ \;\;\;\;\left(0.125 \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right) \cdot \left(D \cdot \frac{M \cdot \left(M \cdot D\right)}{d}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{0.125}{d}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_0 0.0)
     (* (sqrt (/ h (* l (* l l)))) (* (* D_m (* D_m (* M_m M_m))) (/ 0.125 d)))
     (if (<= t_0 2e+136)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (0.125 / d));
	} else if (t_0 <= 2e+136) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (0.125 / d));
	} else if (t_0 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (0.125 / d))
	elif t_0 <= 2e+136:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(0.125 / d)));
	elseif (t_0 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (0.125 / d));
	elseif (t_0 <= 2e+136)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))) < 0.0

   1. Initial program 82.8%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-neg-frac N/A

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

   5. Applied rewrites 34.4%

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

   if 0.0 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 97.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 97.1%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \frac{-0.125 \cdot \left(M\_m \cdot M\_m\right)}{d}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_0 -5e+251)
     (*
      (sqrt (/ h (* l (* l l))))
      (* (* D_m D_m) (/ (* -0.125 (* M_m M_m)) d)))
     (if (<= t_0 2e+136)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* d (sqrt (/ h l))) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e+251) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	} else if (t_0 <= 2e+136) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e+251) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	} else if (t_0 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_0 <= -5e+251:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d))
	elif t_0 <= 2e+136:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -5e+251)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * D_m) * Float64(Float64(-0.125 * Float64(M_m * M_m)) / d)));
	elseif (t_0 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -5e+251)
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	elseif (t_0 <= 2e+136)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+251], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))) < -5.0000000000000005e251

   1. Initial program 84.2%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. associate-*r/ N/A

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

      21. lower-/.f64 N/A

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

   5. Applied rewrites 29.6%

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

   if -5.0000000000000005e251 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 91.9%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 72.8%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -5 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{d \cdot t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -2e-180)
     (/ (* (- d) t_0) h)
     (if (<= t_1 2e+136)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_1 INFINITY)
         (/ (* d t_0) h)
         (* (- d) (sqrt (/ 1.0 (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -2e-180) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 2e+136) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (d * t_0) / h;
	} else {
		tmp = -d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -2e-180) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 2e+136) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (d * t_0) / h;
	} else {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_1 <= -2e-180:
		tmp = (-d * t_0) / h
	elif t_1 <= 2e+136:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_1 <= math.inf:
		tmp = (d * t_0) / h
	else:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -2e-180)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif (t_1 <= 2e+136)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(d * t_0) / h);
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_1 <= -2e-180)
		tmp = (-d * t_0) / h;
	elseif (t_1 <= 2e+136)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_1 <= Inf)
		tmp = (d * t_0) / h;
	else
		tmp = -d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-180], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 2e+136], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(d * t$95$0), $MachinePrecision] / h), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -2e-180

   1. Initial program 86.4%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 22.4%

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

   if -2e-180 < (*.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)))) < 2.00000000000000012e136

   1. Initial program 90.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 90.3%

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

   if 2.00000000000000012e136 < (*.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)))) < +inf.0

   1. Initial program 64.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 80.8%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_3 := \sqrt{\frac{\ell}{d}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)}, \frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot 0.5\right)\right)}{d}, 1\right)}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d} \cdot \frac{M\_m \cdot D\_m}{d \cdot 4}, \frac{h}{\ell} \cdot -0.5, 1\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_1, d \cdot t\_1\right)}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ h l)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_3 (sqrt (/ l d))))
   (if (<= t_2 (- INFINITY))
     (/
      (*
       t_0
       (fma
        (/ (* (* M_m D_m) -0.5) (* l (* d 2.0)))
        (/ (* h (* M_m (* D_m 0.5))) d)
        1.0))
      t_3)
     (if (<= t_2 5e+198)
       (/
        (*
         t_0
         (fma
          (* (/ (* M_m D_m) d) (/ (* M_m D_m) (* d 4.0)))
          (* (/ h l) -0.5)
          1.0))
        t_3)
       (/
        (fma
         (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
         t_1
         (* d t_1))
        h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((h / l));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_3 = sqrt((l / d));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * fma((((M_m * D_m) * -0.5) / (l * (d * 2.0))), ((h * (M_m * (D_m * 0.5))) / d), 1.0)) / t_3;
	} else if (t_2 <= 5e+198) {
		tmp = (t_0 * fma((((M_m * D_m) / d) * ((M_m * D_m) / (d * 4.0))), ((h / l) * -0.5), 1.0)) / t_3;
	} else {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_1, (d * t_1)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(h / l))
	t_2 = 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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_3 = sqrt(Float64(l / d))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / Float64(l * Float64(d * 2.0))), Float64(Float64(h * Float64(M_m * Float64(D_m * 0.5))) / d), 1.0)) / t_3);
	elseif (t_2 <= 5e+198)
		tmp = Float64(Float64(t_0 * fma(Float64(Float64(Float64(M_m * D_m) / d) * Float64(Float64(M_m * D_m) / Float64(d * 4.0))), Float64(Float64(h / l) * -0.5), 1.0)) / t_3);
	else
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_1, Float64(d * t_1)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 5e+198], N[(N[(t$95$0 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(d * t$95$1), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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)))) < -inf.0

   1. Initial program 83.6%

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

   4. Applied rewrites 71.4%

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

   6. Applied rewrites 91.9%

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

   if -inf.0 < (*.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)))) < 5.00000000000000049e198

   1. Initial program 92.6%

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

   4. Applied rewrites 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

   if 5.00000000000000049e198 < (*.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 27.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   7. Applied rewrites 36.4%

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

   9. Applied rewrites 45.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)}, \frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{d}, 1\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d \cdot 4}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(1 - \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_0, d \cdot t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (+
          1.0
          (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
        2e+136)
     (*
      (-
       1.0
       (*
        (/ (/ (* M_m (* D_m 0.5)) (* d 2.0)) l)
        (* h (/ (* 0.5 (* M_m D_m)) d))))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (/
      (fma
       (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
       t_0
       (* d t_0))
      h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 2e+136) {
		tmp = (1.0 - ((((M_m * (D_m * 0.5)) / (d * 2.0)) / l) * (h * ((0.5 * (M_m * D_m)) / d)))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_0, (d * t_0)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (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(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))) <= 2e+136)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d * 2.0)) / l) * Float64(h * Float64(Float64(0.5 * Float64(M_m * D_m)) / d)))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_0, Float64(d * t_0)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+136], N[(N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(d * t$95$0), $MachinePrecision]), $MachinePrecision] / h), $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)))) < 2.00000000000000012e136

   1. Initial program 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.8

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

   6. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 92.8

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 92.8

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 92.8

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

   8. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 92.8

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

   10. Applied rewrites 92.8%

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

   if 2.00000000000000012e136 < (*.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 32.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 20.7%

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

   7. Applied rewrites 39.3%

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

   9. Applied rewrites 48.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := 1 - \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_0, d \cdot t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (-
          1.0
          (*
           (/ (/ (* M_m (* D_m 0.5)) (* d 2.0)) l)
           (* h (/ (* 0.5 (* M_m D_m)) d))))))
   (if (<= d -8.2e-193)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))) t_1)
     (if (<= d 8e-289)
       (/
        (fma
         (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
         t_0
         (* d t_0))
        h)
       (* t_1 (* (pow (/ d h) (/ 1.0 2.0)) (/ (sqrt d) (sqrt l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = 1.0 - ((((M_m * (D_m * 0.5)) / (d * 2.0)) / l) * (h * ((0.5 * (M_m * D_m)) / d)));
	double tmp;
	if (d <= -8.2e-193) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * t_1;
	} else if (d <= 8e-289) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_0, (d * t_0)) / h;
	} else {
		tmp = t_1 * (pow((d / h), (1.0 / 2.0)) * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d * 2.0)) / l) * Float64(h * Float64(Float64(0.5 * Float64(M_m * D_m)) / d))))
	tmp = 0.0
	if (d <= -8.2e-193)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_1);
	elseif (d <= 8e-289)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_0, Float64(d * t_0)) / h);
	else
		tmp = Float64(t_1 * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.2e-193], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 8e-289], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(d * t$95$0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$1 * N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.20000000000000005e-193

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 80.1

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 80.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 91.4

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

   10. Applied rewrites 91.4%

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

   if -8.20000000000000005e-193 < d < 8.0000000000000001e-289

   1. Initial program 31.1%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 59.8%

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

   if 8.0000000000000001e-289 < d

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 79.4

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 79.4

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 79.4

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 79.4

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

   8. Applied rewrites 79.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 85.9

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

   10. Applied rewrites 85.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+156}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-188}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_2, d \cdot t\_2\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (*
           (/ (/ (* M_m (* D_m 0.5)) (* d 2.0)) l)
           (* h (/ (* 0.5 (* M_m D_m)) d)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ h l))))
   (if (<= d -6.2e+156)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -6.6e-188)
       (* t_0 (* t_1 (sqrt (/ d h))))
       (if (<= d 2.5e-237)
         (/
          (fma
           (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
           t_2
           (* d t_2))
          h)
         (* t_0 (* t_1 (/ (sqrt d) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - ((((M_m * (D_m * 0.5)) / (d * 2.0)) / l) * (h * ((0.5 * (M_m * D_m)) / d)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / l));
	double tmp;
	if (d <= -6.2e+156) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -6.6e-188) {
		tmp = t_0 * (t_1 * sqrt((d / h)));
	} else if (d <= 2.5e-237) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_2, (d * t_2)) / h;
	} else {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d * 2.0)) / l) * Float64(h * Float64(Float64(0.5 * Float64(M_m * D_m)) / d))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -6.2e+156)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -6.6e-188)
		tmp = Float64(t_0 * Float64(t_1 * sqrt(Float64(d / h))));
	elseif (d <= 2.5e-237)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_2, Float64(d * t_2)) / h);
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.2e+156], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.6e-188], N[(t$95$0 * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e-237], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(d * t$95$2), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.2000000000000004e156

   1. Initial program 61.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -6.2000000000000004e156 < d < -6.6000000000000005e-188

   1. Initial program 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 87.4

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

   6. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 87.4

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 87.4

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 87.4

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

   8. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -6.6000000000000005e-188 < d < 2.5000000000000001e-237

   1. Initial program 36.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 18.9%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 54.2%

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

   if 2.5000000000000001e-237 < d

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 85.1

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

   6. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 85.1

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 85.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 85.1

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

   8. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 85.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 92.0

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

   10. Applied rewrites 92.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+156}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-188}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)}, \frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot 0.5\right)\right)}{d}, 1\right)}{\sqrt{\frac{\ell}{d}}}\\ t_1 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{+149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_1, d \cdot t\_1\right)}{h}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (sqrt (/ d h))
           (fma
            (/ (* (* M_m D_m) -0.5) (* l (* d 2.0)))
            (/ (* h (* M_m (* D_m 0.5))) d)
            1.0))
          (sqrt (/ l d))))
        (t_1 (sqrt (/ h l))))
   (if (<= d -3.7e+149)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -6.8e-132)
       t_0
       (if (<= d 4.4e-231)
         (/
          (fma
           (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
           t_1
           (* d t_1))
          h)
         (if (<= d 3.2e+224) t_0 (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (sqrt((d / h)) * fma((((M_m * D_m) * -0.5) / (l * (d * 2.0))), ((h * (M_m * (D_m * 0.5))) / d), 1.0)) / sqrt((l / d));
	double t_1 = sqrt((h / l));
	double tmp;
	if (d <= -3.7e+149) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -6.8e-132) {
		tmp = t_0;
	} else if (d <= 4.4e-231) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_1, (d * t_1)) / h;
	} else if (d <= 3.2e+224) {
		tmp = t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(sqrt(Float64(d / h)) * fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / Float64(l * Float64(d * 2.0))), Float64(Float64(h * Float64(M_m * Float64(D_m * 0.5))) / d), 1.0)) / sqrt(Float64(l / d)))
	t_1 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -3.7e+149)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -6.8e-132)
		tmp = t_0;
	elseif (d <= 4.4e-231)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_1, Float64(d * t_1)) / h);
	elseif (d <= 3.2e+224)
		tmp = t_0;
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.7e+149], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.8e-132], t$95$0, If[LessEqual[d, 4.4e-231], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(d * t$95$1), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 3.2e+224], t$95$0, N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.69999999999999978e149

   1. Initial program 61.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -3.69999999999999978e149 < d < -6.79999999999999965e-132 or 4.40000000000000018e-231 < d < 3.20000000000000015e224

   1. Initial program 79.9%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 84.9%

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

   if -6.79999999999999965e-132 < d < 4.40000000000000018e-231

   1. Initial program 40.1%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 21.6%

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

   7. Applied rewrites 46.3%

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

   9. Applied rewrites 54.4%

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

   if 3.20000000000000015e224 < d

   1. Initial program 61.7%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 86.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{+149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)}, \frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{d}, 1\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)}, \frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{d}, 1\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\right)\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_0, d \cdot t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (-
          1.0
          (*
           (/ (/ (* M_m (* D_m 0.5)) (* d 2.0)) l)
           (* h (/ (* 0.5 (* M_m D_m)) d))))))
   (if (<= d -8.2e-193)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_1) t_2)
     (if (<= d 2.5e-237)
       (/
        (fma
         (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
         t_0
         (* d t_0))
        h)
       (* t_2 (* t_1 (/ (sqrt d) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 - ((((M_m * (D_m * 0.5)) / (d * 2.0)) / l) * (h * ((0.5 * (M_m * D_m)) / d)));
	double tmp;
	if (d <= -8.2e-193) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_1) * t_2;
	} else if (d <= 2.5e-237) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_0, (d * t_0)) / h;
	} else {
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d * 2.0)) / l) * Float64(h * Float64(Float64(0.5 * Float64(M_m * D_m)) / d))))
	tmp = 0.0
	if (d <= -8.2e-193)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_1) * t_2);
	elseif (d <= 2.5e-237)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_0, Float64(d * t_0)) / h);
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.2e-193], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 2.5e-237], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(d * t$95$0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.20000000000000005e-193

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lower-*.f64 80.1

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 80.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 91.4

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

   10. Applied rewrites 91.4%

       \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

   if -8.20000000000000005e-193 < d < 2.5000000000000001e-237

   1. Initial program 36.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.9%

      \[\leadsto \frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{3}\right)\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 2.5000000000000001e-237 < d

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      7. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}}{\frac{\ell}{h}}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}}{\frac{\ell}{h}}\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 85.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 85.1

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 85.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}}}\right) \]

      2. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{1}{\frac{1}{h}}\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{1}{\color{blue}{\frac{1}{h}}}\right)\right) \]

      4. remove-double-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{h}\right)\right) \]

      5. lower-*.f64 85.1

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot h\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot h\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot h\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot h\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot h\right)\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot h\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}}{d} \cdot h\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot \left(D \cdot \frac{1}{2}\right)}}{d} \cdot h\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{M \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}}{d} \cdot h\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot \left(D \cdot \frac{1}{2}\right)}}{d} \cdot h\right)\right) \]

      15. lower-/.f64 85.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{M \cdot \left(D \cdot 0.5\right)}{d}} \cdot h\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot \left(D \cdot \frac{1}{2}\right)}}{d} \cdot h\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{M \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}}{d} \cdot h\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot h\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d} \cdot h\right)\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d} \cdot h\right)\right) \]

      21. lift-*.f64 85.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \color{blue}{\left(M \cdot D\right)}}{d} \cdot h\right)\right) \]

   8. Applied rewrites 85.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      2. metadata-eval 85.1

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot \frac{1}{2}\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

      9. lower-sqrt.f64 92.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]

   10. Applied rewrites 92.0%

       \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \left(h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := M\_m \cdot \left(D\_m \cdot 0.5\right)\\ t_2 := \frac{t\_0 \cdot \left(1 - \frac{t\_1 \cdot \left(h \cdot t\_1\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ t_3 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_3, d \cdot t\_3\right)}{h}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right) \cdot \left(1 - \left(\frac{D\_m}{d} \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)\right) \cdot \left(\frac{D\_m}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (* M_m (* D_m 0.5)))
        (t_2
         (/
          (* t_0 (- 1.0 (/ (* t_1 (* h t_1)) (* d (* l (* d 2.0))))))
          (sqrt (/ l d))))
        (t_3 (sqrt (/ h l))))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -7.3e-131)
       t_2
       (if (<= d 6.6e-161)
         (/
          (fma
           (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
           t_3
           (* d t_3))
          h)
         (if (<= d 4.8e+130)
           t_2
           (*
            (* (sqrt (/ d l)) t_0)
            (-
             1.0
             (*
              (* (/ D_m d) (* (* M_m M_m) 0.25))
              (* (/ D_m d) (* 0.5 (/ h l))))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = M_m * (D_m * 0.5);
	double t_2 = (t_0 * (1.0 - ((t_1 * (h * t_1)) / (d * (l * (d * 2.0)))))) / sqrt((l / d));
	double t_3 = sqrt((h / l));
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -7.3e-131) {
		tmp = t_2;
	} else if (d <= 6.6e-161) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_3, (d * t_3)) / h;
	} else if (d <= 4.8e+130) {
		tmp = t_2;
	} else {
		tmp = (sqrt((d / l)) * t_0) * (1.0 - (((D_m / d) * ((M_m * M_m) * 0.25)) * ((D_m / d) * (0.5 * (h / l)))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(M_m * Float64(D_m * 0.5))
	t_2 = Float64(Float64(t_0 * Float64(1.0 - Float64(Float64(t_1 * Float64(h * t_1)) / Float64(d * Float64(l * Float64(d * 2.0)))))) / sqrt(Float64(l / d)))
	t_3 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -7.3e-131)
		tmp = t_2;
	elseif (d <= 6.6e-161)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_3, Float64(d * t_3)) / h);
	elseif (d <= 4.8e+130)
		tmp = t_2;
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_0) * Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(Float64(M_m * M_m) * 0.25)) * Float64(Float64(D_m / d) * Float64(0.5 * Float64(h / l))))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(1.0 - N[(N[(t$95$1 * N[(h * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.3e-131], t$95$2, If[LessEqual[d, 6.6e-161], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(d * t$95$3), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 4.8e+130], t$95$2, N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -7.3000000000000001e-131 or 6.5999999999999997e-161 < d < 4.80000000000000048e130

   1. Initial program 82.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 85.3%

      \[\leadsto \frac{\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot d}\right)}}{\sqrt{\frac{\ell}{d}}} \]

   if -7.3000000000000001e-131 < d < 6.5999999999999997e-161

   1. Initial program 38.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 18.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.0%

      \[\leadsto \frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{3}\right)\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 4.80000000000000048e130 < d

   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 rewrites 72.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lower-sqrt.f64 72.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 72.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lower-sqrt.f64 72.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 72.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot 0.5\right)\\ t_1 := \frac{\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{t\_0 \cdot \left(h \cdot t\_0\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_2, d \cdot t\_2\right)}{h}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m 0.5)))
        (t_1
         (/
          (*
           (sqrt (/ d h))
           (- 1.0 (/ (* t_0 (* h t_0)) (* d (* l (* d 2.0))))))
          (sqrt (/ l d))))
        (t_2 (sqrt (/ h l))))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -7.3e-131)
       t_1
       (if (<= d 6.6e-161)
         (/
          (fma
           (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
           t_2
           (* d t_2))
          h)
         (if (<= d 2.1e+191) t_1 (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * 0.5);
	double t_1 = (sqrt((d / h)) * (1.0 - ((t_0 * (h * t_0)) / (d * (l * (d * 2.0)))))) / sqrt((l / d));
	double t_2 = sqrt((h / l));
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -7.3e-131) {
		tmp = t_1;
	} else if (d <= 6.6e-161) {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_2, (d * t_2)) / h;
	} else if (d <= 2.1e+191) {
		tmp = t_1;
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m * 0.5))
	t_1 = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64(t_0 * Float64(h * t_0)) / Float64(d * Float64(l * Float64(d * 2.0)))))) / sqrt(Float64(l / d)))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -7.3e-131)
		tmp = t_1;
	elseif (d <= 6.6e-161)
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_2, Float64(d * t_2)) / h);
	elseif (d <= 2.1e+191)
		tmp = t_1;
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.3e-131], t$95$1, If[LessEqual[d, 6.6e-161], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(d * t$95$2), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 2.1e+191], t$95$1, N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -7.3000000000000001e-131 or 6.5999999999999997e-161 < d < 2.1000000000000001e191

   1. Initial program 83.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 85.0%

      \[\leadsto \frac{\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot d}\right)}}{\sqrt{\frac{\ell}{d}}} \]

   if -7.3000000000000001e-131 < d < 6.5999999999999997e-161

   1. Initial program 38.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 18.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.0%

      \[\leadsto \frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{3}\right)\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 2.1000000000000001e191 < d

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 66.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)\right)}{d \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(d \cdot 4\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := D\_m \cdot \left(M\_m \cdot D\_m\right)\\ t_4 := M\_m \cdot t\_3\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \mathsf{fma}\left(0.5, \frac{t\_3 \cdot \left(h \cdot M\_m\right)}{t\_0 \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{t\_4 \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t\_4}{d} \cdot \left(h \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right), -0.125, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(1 + \frac{t\_4 \cdot \left(h \cdot -0.5\right)}{\ell \cdot t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (* d 4.0)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (* D_m (* M_m D_m)))
        (t_4 (* M_m t_3)))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -5.7e-133)
       (* t_1 (* t_2 (fma 0.5 (/ (* t_3 (* h M_m)) (* t_0 (- l))) 1.0)))
       (if (<= d 9.2e-294)
         (* (sqrt (/ (/ h (* l l)) l)) (/ (* t_4 0.125) d))
         (if (<= d 1.8e-189)
           (/
            (fma
             (* (/ t_4 d) (* h (sqrt (/ h (* l (* l l))))))
             -0.125
             (* d (sqrt (/ h l))))
            h)
           (if (<= d 3e+108)
             (* t_2 (* t_1 (+ 1.0 (/ (* t_4 (* h -0.5)) (* l t_0)))))
             (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * (d * 4.0);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = D_m * (M_m * D_m);
	double t_4 = M_m * t_3;
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -5.7e-133) {
		tmp = t_1 * (t_2 * fma(0.5, ((t_3 * (h * M_m)) / (t_0 * -l)), 1.0));
	} else if (d <= 9.2e-294) {
		tmp = sqrt(((h / (l * l)) / l)) * ((t_4 * 0.125) / d);
	} else if (d <= 1.8e-189) {
		tmp = fma(((t_4 / d) * (h * sqrt((h / (l * (l * l)))))), -0.125, (d * sqrt((h / l)))) / h;
	} else if (d <= 3e+108) {
		tmp = t_2 * (t_1 * (1.0 + ((t_4 * (h * -0.5)) / (l * t_0))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * Float64(d * 4.0))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(D_m * Float64(M_m * D_m))
	t_4 = Float64(M_m * t_3)
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -5.7e-133)
		tmp = Float64(t_1 * Float64(t_2 * fma(0.5, Float64(Float64(t_3 * Float64(h * M_m)) / Float64(t_0 * Float64(-l))), 1.0)));
	elseif (d <= 9.2e-294)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(t_4 * 0.125) / d));
	elseif (d <= 1.8e-189)
		tmp = Float64(fma(Float64(Float64(t_4 / d) * Float64(h * sqrt(Float64(h / Float64(l * Float64(l * l)))))), -0.125, Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (d <= 3e+108)
		tmp = Float64(t_2 * Float64(t_1 * Float64(1.0 + Float64(Float64(t_4 * Float64(h * -0.5)) / Float64(l * t_0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M$95$m * t$95$3), $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.7e-133], N[(t$95$1 * N[(t$95$2 * N[(0.5 * N[(N[(t$95$3 * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * (-l)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-294], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$4 * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-189], N[(N[(N[(N[(t$95$4 / d), $MachinePrecision] * N[(h * N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 3e+108], N[(t$95$2 * N[(t$95$1 * N[(1.0 + N[(N[(t$95$4 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -5.6999999999999997e-133

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   6. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot \left(-h\right)\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -5.6999999999999997e-133 < d < 9.20000000000000064e-294

   1. Initial program 37.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right)} \]

   7. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.8%

      \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \left(\left(\color{blue}{0.125} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{0.125 \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{d}} \]

   if 9.20000000000000064e-294 < d < 1.80000000000000008e-189

   1. Initial program 41.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 24.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d} \cdot \left(h \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right), -0.125, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 1.80000000000000008e-189 < d < 2.99999999999999984e108

   1. Initial program 77.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if 2.99999999999999984e108 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 60.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot M\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d} \cdot \left(h \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right), -0.125, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 63.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(d \cdot 4\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := D\_m \cdot \left(M\_m \cdot D\_m\right)\\ t_4 := M\_m \cdot t\_3\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \mathsf{fma}\left(0.5, \frac{t\_3 \cdot \left(h \cdot M\_m\right)}{t\_0 \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{t\_4 \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \frac{-0.125 \cdot \left(M\_m \cdot M\_m\right)}{d}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(1 + \frac{t\_4 \cdot \left(h \cdot -0.5\right)}{\ell \cdot t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (* d 4.0)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (* D_m (* M_m D_m)))
        (t_4 (* M_m t_3)))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -5.7e-133)
       (* t_1 (* t_2 (fma 0.5 (/ (* t_3 (* h M_m)) (* t_0 (- l))) 1.0)))
       (if (<= d 9.2e-302)
         (* (sqrt (/ (/ h (* l l)) l)) (/ (* t_4 0.125) d))
         (if (<= d 1.15e-231)
           (*
            (sqrt (/ h (* l (* l l))))
            (* (* D_m D_m) (/ (* -0.125 (* M_m M_m)) d)))
           (if (<= d 3e+108)
             (* t_2 (* t_1 (+ 1.0 (/ (* t_4 (* h -0.5)) (* l t_0)))))
             (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * (d * 4.0);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = D_m * (M_m * D_m);
	double t_4 = M_m * t_3;
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -5.7e-133) {
		tmp = t_1 * (t_2 * fma(0.5, ((t_3 * (h * M_m)) / (t_0 * -l)), 1.0));
	} else if (d <= 9.2e-302) {
		tmp = sqrt(((h / (l * l)) / l)) * ((t_4 * 0.125) / d);
	} else if (d <= 1.15e-231) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	} else if (d <= 3e+108) {
		tmp = t_2 * (t_1 * (1.0 + ((t_4 * (h * -0.5)) / (l * t_0))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * Float64(d * 4.0))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(D_m * Float64(M_m * D_m))
	t_4 = Float64(M_m * t_3)
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -5.7e-133)
		tmp = Float64(t_1 * Float64(t_2 * fma(0.5, Float64(Float64(t_3 * Float64(h * M_m)) / Float64(t_0 * Float64(-l))), 1.0)));
	elseif (d <= 9.2e-302)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(t_4 * 0.125) / d));
	elseif (d <= 1.15e-231)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * D_m) * Float64(Float64(-0.125 * Float64(M_m * M_m)) / d)));
	elseif (d <= 3e+108)
		tmp = Float64(t_2 * Float64(t_1 * Float64(1.0 + Float64(Float64(t_4 * Float64(h * -0.5)) / Float64(l * t_0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M$95$m * t$95$3), $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.7e-133], N[(t$95$1 * N[(t$95$2 * N[(0.5 * N[(N[(t$95$3 * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * (-l)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-302], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$4 * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-231], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+108], N[(t$95$2 * N[(t$95$1 * N[(1.0 + N[(N[(t$95$4 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -5.6999999999999997e-133

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   6. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot \left(-h\right)\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -5.6999999999999997e-133 < d < 9.20000000000000007e-302

   1. Initial program 38.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right)} \]

   7. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 31.5%

      \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \left(\left(\color{blue}{0.125} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.1%

       \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{0.125 \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{d}} \]

   if 9.20000000000000007e-302 < d < 1.15e-231

   1. Initial program 43.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot {D}^{2}\right)}}{d} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot {D}^{2}}}{d} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)}}{d} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{\frac{-1}{8} \cdot {M}^{2}}{d}\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      18. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      20. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{d}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{d}}\right) \]

   5. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}\right)} \]

   if 1.15e-231 < d < 2.99999999999999984e108

   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 rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if 2.99999999999999984e108 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 60.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot M\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 63.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{t\_0 \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{t\_0 \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \frac{-0.125 \cdot \left(M\_m \cdot M\_m\right)}{d}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (* M_m D_m))))
        (t_1
         (*
          (sqrt (/ d l))
          (*
           (sqrt (/ d h))
           (+ 1.0 (/ (* t_0 (* h -0.5)) (* l (* d (* d 4.0)))))))))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -5.7e-133)
       t_1
       (if (<= d 9.2e-302)
         (* (sqrt (/ (/ h (* l l)) l)) (/ (* t_0 0.125) d))
         (if (<= d 1.15e-231)
           (*
            (sqrt (/ h (* l (* l l))))
            (* (* D_m D_m) (/ (* -0.125 (* M_m M_m)) d)))
           (if (<= d 3e+108) t_1 (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0))))));
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -5.7e-133) {
		tmp = t_1;
	} else if (d <= 9.2e-302) {
		tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	} else if (d <= 1.15e-231) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	} else if (d <= 3e+108) {
		tmp = t_1;
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m_m * (d_m * (m_m * d_m))
    t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((t_0 * (h * (-0.5d0))) / (l * (d * (d * 4.0d0))))))
    if (d <= (-1.66d+146)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (d <= (-5.7d-133)) then
        tmp = t_1
    else if (d <= 9.2d-302) then
        tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125d0) / d)
    else if (d <= 1.15d-231) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_m * d_m) * (((-0.125d0) * (m_m * m_m)) / d))
    else if (d <= 3d+108) then
        tmp = t_1
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double t_1 = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0))))));
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (d <= -5.7e-133) {
		tmp = t_1;
	} else if (d <= 9.2e-302) {
		tmp = Math.sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	} else if (d <= 1.15e-231) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	} else if (d <= 3e+108) {
		tmp = t_1;
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * (D_m * (M_m * D_m))
	t_1 = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0))))))
	tmp = 0
	if d <= -1.66e+146:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif d <= -5.7e-133:
		tmp = t_1
	elif d <= 9.2e-302:
		tmp = math.sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d)
	elif d <= 1.15e-231:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d))
	elif d <= 3e+108:
		tmp = t_1
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(M_m * D_m)))
	t_1 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(t_0 * Float64(h * -0.5)) / Float64(l * Float64(d * Float64(d * 4.0)))))))
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -5.7e-133)
		tmp = t_1;
	elseif (d <= 9.2e-302)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(t_0 * 0.125) / d));
	elseif (d <= 1.15e-231)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * D_m) * Float64(Float64(-0.125 * Float64(M_m * M_m)) / d)));
	elseif (d <= 3e+108)
		tmp = t_1;
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * (D_m * (M_m * D_m));
	t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0))))));
	tmp = 0.0;
	if (d <= -1.66e+146)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (d <= -5.7e-133)
		tmp = t_1;
	elseif (d <= 9.2e-302)
		tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	elseif (d <= 1.15e-231)
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * D_m) * ((-0.125 * (M_m * M_m)) / d));
	elseif (d <= 3e+108)
		tmp = t_1;
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(t$95$0 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.7e-133], t$95$1, If[LessEqual[d, 9.2e-302], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-231], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+108], t$95$1, N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -5.6999999999999997e-133 or 1.15e-231 < d < 2.99999999999999984e108

   1. Initial program 78.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -5.6999999999999997e-133 < d < 9.20000000000000007e-302

   1. Initial program 38.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right)} \]

   7. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 31.5%

      \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \left(\left(\color{blue}{0.125} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.1%

       \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{0.125 \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{d}} \]

   if 9.20000000000000007e-302 < d < 1.15e-231

   1. Initial program 43.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot {D}^{2}\right)}}{d} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot {D}^{2}}}{d} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)}}{d} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{\frac{-1}{8} \cdot {M}^{2}}{d}\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      18. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      20. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{d}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{d}}\right) \]

   5. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}\right)} \]

   if 2.99999999999999984e108 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 60.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 65.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(h \cdot M\_m\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, t\_0, d \cdot t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= d -1.66e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -7.3e-131)
       (*
        (sqrt (/ d h))
        (*
         (sqrt (/ d l))
         (fma
          0.5
          (/ (* (* D_m (* M_m D_m)) (* h M_m)) (* (* d (* d 4.0)) (- l)))
          1.0)))
       (/
        (fma
         (* (* D_m (* (/ (* M_m (* M_m D_m)) d) -0.125)) (/ h l))
         t_0
         (* d t_0))
        h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (d <= -1.66e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -7.3e-131) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma(0.5, (((D_m * (M_m * D_m)) * (h * M_m)) / ((d * (d * 4.0)) * -l)), 1.0));
	} else {
		tmp = fma(((D_m * (((M_m * (M_m * D_m)) / d) * -0.125)) * (h / l)), t_0, (d * t_0)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -1.66e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -7.3e-131)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(0.5, Float64(Float64(Float64(D_m * Float64(M_m * D_m)) * Float64(h * M_m)) / Float64(Float64(d * Float64(d * 4.0)) * Float64(-l))), 1.0)));
	else
		tmp = Float64(fma(Float64(Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / d) * -0.125)) * Float64(h / l)), t_0, Float64(d * t_0)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.66e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.3e-131], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(N[(N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(d * t$95$0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6600000000000001e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.6600000000000001e146 < d < -7.3000000000000001e-131

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   6. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot \left(-h\right)\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -7.3000000000000001e-131 < d

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot \left(-0.125 \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{3}\right)\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(0.5, \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot M\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \left(-\ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot \left(\frac{M \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\right) \cdot \frac{h}{\ell}, \sqrt{\frac{h}{\ell}}, d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{t\_0 \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{t\_0 \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (* M_m D_m)))))
   (if (<= d -1.65e+146)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d -2.15e-156)
       (*
        (fma (/ t_0 (* 4.0 (* d d))) (* (/ h l) -0.5) 1.0)
        (sqrt (/ (* d d) (* h l))))
       (if (<= d 3.5e-246)
         (* (sqrt (/ (/ h (* l l)) l)) (/ (* t_0 0.125) d))
         (if (<= d 3e+113)
           (/
            (* d (+ 1.0 (/ (* t_0 (* h -0.5)) (* l (* d (* d 4.0))))))
            (sqrt (* h l)))
           (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double tmp;
	if (d <= -1.65e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -2.15e-156) {
		tmp = fma((t_0 / (4.0 * (d * d))), ((h / l) * -0.5), 1.0) * sqrt(((d * d) / (h * l)));
	} else if (d <= 3.5e-246) {
		tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	} else if (d <= 3e+113) {
		tmp = (d * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0)))))) / sqrt((h * l));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(M_m * D_m)))
	tmp = 0.0
	if (d <= -1.65e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -2.15e-156)
		tmp = Float64(fma(Float64(t_0 / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (d <= 3.5e-246)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(t_0 * 0.125) / d));
	elseif (d <= 3e+113)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(Float64(t_0 * Float64(h * -0.5)) / Float64(l * Float64(d * Float64(d * 4.0)))))) / sqrt(Float64(h * l)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.65e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.15e-156], N[(N[(N[(t$95$0 / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-246], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+113], N[(N[(d * N[(1.0 + N[(N[(t$95$0 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.65000000000000008e146

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 77.1

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.65000000000000008e146 < d < -2.14999999999999989e-156

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

   if -2.14999999999999989e-156 < d < 3.5000000000000002e-246

   1. Initial program 37.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right)} \]

   7. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.5%

      \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \left(\left(\color{blue}{0.125} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 41.5%

       \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{0.125 \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{d}} \]

   if 3.5000000000000002e-246 < d < 3e113

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}} \]

   if 3e113 < d

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 59.2

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 57.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+45}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{t\_0 \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{t\_0 \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (* M_m D_m)))))
   (if (<= d -1.9e+45)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= d 3.5e-246)
       (* (sqrt (/ (/ h (* l l)) l)) (/ (* t_0 0.125) d))
       (if (<= d 3e+113)
         (/
          (* d (+ 1.0 (/ (* t_0 (* h -0.5)) (* l (* d (* d 4.0))))))
          (sqrt (* h l)))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double tmp;
	if (d <= -1.9e+45) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= 3.5e-246) {
		tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	} else if (d <= 3e+113) {
		tmp = (d * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0)))))) / sqrt((h * l));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_m * (m_m * d_m))
    if (d <= (-1.9d+45)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (d <= 3.5d-246) then
        tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125d0) / d)
    else if (d <= 3d+113) then
        tmp = (d * (1.0d0 + ((t_0 * (h * (-0.5d0))) / (l * (d * (d * 4.0d0)))))) / sqrt((h * l))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double tmp;
	if (d <= -1.9e+45) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (d <= 3.5e-246) {
		tmp = Math.sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	} else if (d <= 3e+113) {
		tmp = (d * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0)))))) / Math.sqrt((h * l));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * (D_m * (M_m * D_m))
	tmp = 0
	if d <= -1.9e+45:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif d <= 3.5e-246:
		tmp = math.sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d)
	elif d <= 3e+113:
		tmp = (d * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0)))))) / math.sqrt((h * l))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(M_m * D_m)))
	tmp = 0.0
	if (d <= -1.9e+45)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= 3.5e-246)
		tmp = Float64(sqrt(Float64(Float64(h / Float64(l * l)) / l)) * Float64(Float64(t_0 * 0.125) / d));
	elseif (d <= 3e+113)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(Float64(t_0 * Float64(h * -0.5)) / Float64(l * Float64(d * Float64(d * 4.0)))))) / sqrt(Float64(h * l)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * (D_m * (M_m * D_m));
	tmp = 0.0;
	if (d <= -1.9e+45)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (d <= 3.5e-246)
		tmp = sqrt(((h / (l * l)) / l)) * ((t_0 * 0.125) / d);
	elseif (d <= 3e+113)
		tmp = (d * (1.0 + ((t_0 * (h * -0.5)) / (l * (d * (d * 4.0)))))) / sqrt((h * l));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.9e+45], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-246], N[(N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * 0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+113], N[(N[(d * N[(1.0 + N[(N[(t$95$0 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.9000000000000001e45

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 68.0

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.9000000000000001e45 < d < 3.5000000000000002e-246

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 26.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot 0.5, \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot \left(-h\right)}{\ell}, \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right)} \]

   7. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 31.6%

      \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \left(\left(\color{blue}{0.125} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 41.9%

       \[\leadsto \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{0.125 \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{d}} \]

   if 3.5000000000000002e-246 < d < 3e113

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{\ell}{d}}}} \]

   6. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}} \]

   if 3e113 < d

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 59.2

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto d \cdot \frac{\sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+45}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}} \cdot \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot 0.125}{d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 46.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-262}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 5.5e-262)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.5e-262) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 5.5d-262) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.5e-262) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 5.5e-262:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 5.5e-262)
		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

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 5.5e-262)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 5.5e-262], 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 < 5.5000000000000004e-262

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 38.7

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 5.5000000000000004e-262 < l

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 38.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 38.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-262}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 43.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-193}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.95e-193) (* (- d) (sqrt (/ 1.0 (* h l)))) (/ d (sqrt (* h l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.95e-193) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.95d-193)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.95e-193) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.95e-193:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.95e-193)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.95e-193)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.95e-193], 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 d < -1.9499999999999999e-193

   1. Initial program 73.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 47.0

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.9499999999999999e-193 < d

   1. Initial program 62.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 32.5

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 32.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-193}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 37.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -4.5e-155) (sqrt (* d (/ d (* h l)))) (/ d (sqrt (* h l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.5e-155) {
		tmp = sqrt((d * (d / (h * l))));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-4.5d-155)) then
        tmp = sqrt((d * (d / (h * l))))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.5e-155) {
		tmp = Math.sqrt((d * (d / (h * l))));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -4.5e-155:
		tmp = math.sqrt((d * (d / (h * l))))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -4.5e-155)
		tmp = sqrt(Float64(d * Float64(d / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -4.5e-155)
		tmp = sqrt((d * (d / (h * l))));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -4.5e-155], N[Sqrt[N[(d * N[(d / 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 d < -4.5000000000000004e-155

   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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 5.0

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 5.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.7%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 33.5%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \]

   if -4.5000000000000004e-155 < d

   1. Initial program 62.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. lower-*.f64 31.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 31.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 26.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. 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. lower-*.f64 21.2

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

5. Applied rewrites 21.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 21.3%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(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)))))