Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.3% → 78.5%

Time: 17.3s

Alternatives: 18

Speedup: 2.8×

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: 67.3% 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: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \frac{M \cdot D}{d}\\ t_2 := \frac{D \cdot M}{d} \cdot 0.5\\ t_3 := \left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{t\_2}{-1} \cdot \frac{t\_2 \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot t\_1\right), h \cdot \frac{0.25 \cdot t\_1}{\ell}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1 (/ (* M D) d))
        (t_2 (* (/ (* D M) d) 0.5))
        (t_3
         (*
          (* (pow (sqrt (/ h d)) -1.0) (sqrt (/ d l)))
          (fma (* (/ t_2 -1.0) (/ (* t_2 0.5) l)) h 1.0))))
   (if (<= d -2.1e+107)
     (*
      (* (- d) (sqrt (pow (* l h) -1.0)))
      (fma (/ (* -0.5 (pow (* (/ d M) (/ 2.0 D)) -2.0)) l) h 1.0))
     (if (<= d -1.02e-47)
       t_3
       (if (<= d 1.1e-272)
         (/
          (* (sqrt (/ h l)) (fma (/ h l) (/ (* -0.125 (pow (* M D) 2.0)) d) d))
          h)
         (if (<= d 6.3e-86)
           (fma (* t_0 (* -0.5 t_1)) (* h (/ (* 0.25 t_1) l)) t_0)
           t_3))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = (M * D) / d;
	double t_2 = ((D * M) / d) * 0.5;
	double t_3 = (pow(sqrt((h / d)), -1.0) * sqrt((d / l))) * fma(((t_2 / -1.0) * ((t_2 * 0.5) / l)), h, 1.0);
	double tmp;
	if (d <= -2.1e+107) {
		tmp = (-d * sqrt(pow((l * h), -1.0))) * fma(((-0.5 * pow(((d / M) * (2.0 / D)), -2.0)) / l), h, 1.0);
	} else if (d <= -1.02e-47) {
		tmp = t_3;
	} else if (d <= 1.1e-272) {
		tmp = (sqrt((h / l)) * fma((h / l), ((-0.125 * pow((M * D), 2.0)) / d), d)) / h;
	} else if (d <= 6.3e-86) {
		tmp = fma((t_0 * (-0.5 * t_1)), (h * ((0.25 * t_1) / l)), t_0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = Float64(Float64(M * D) / d)
	t_2 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_3 = Float64(Float64((sqrt(Float64(h / d)) ^ -1.0) * sqrt(Float64(d / l))) * fma(Float64(Float64(t_2 / -1.0) * Float64(Float64(t_2 * 0.5) / l)), h, 1.0))
	tmp = 0.0
	if (d <= -2.1e+107)
		tmp = Float64(Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0))) * fma(Float64(Float64(-0.5 * (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0)) / l), h, 1.0));
	elseif (d <= -1.02e-47)
		tmp = t_3;
	elseif (d <= 1.1e-272)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(h / l), Float64(Float64(-0.125 * (Float64(M * D) ^ 2.0)) / d), d)) / h);
	elseif (d <= 6.3e-86)
		tmp = fma(Float64(t_0 * Float64(-0.5 * t_1)), Float64(h * Float64(Float64(0.25 * t_1) / l)), t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$2 / -1.0), $MachinePrecision] * N[(N[(t$95$2 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.1e+107], N[(N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.02e-47], t$95$3, If[LessEqual[d, 1.1e-272], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 6.3e-86], N[(N[(t$95$0 * N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.1e107

   1. Initial program 67.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 77.9

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

   6. Applied rewrites 77.9%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.2

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

   9. Applied rewrites 92.2%

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

   if -2.1e107 < d < -1.02000000000000002e-47 or 6.2999999999999999e-86 < d

   1. Initial program 81.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.2

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

   6. Applied rewrites 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 88.7

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

   10. Applied rewrites 88.7%

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

   if -1.02000000000000002e-47 < d < 1.09999999999999994e-272

   1. Initial program 46.8%

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

   3. Taylor expanded in h around 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 44.7%

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

   7. Applied rewrites 50.7%

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

   9. Applied rewrites 75.7%

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

   if 1.09999999999999994e-272 < d < 6.2999999999999999e-86

   1. Initial program 37.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 37.6

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

   6. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 76.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d} \cdot 0.5\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{h}{\ell}}\\ t_3 := t\_2 \cdot d\\ t_4 := \sqrt{\frac{d}{\ell}}\\ t_5 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot t\_4\right) \cdot t\_5\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;t\_4 \cdot t\_5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_3}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot t\_2, t\_3\right)}{h}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (* D M) d) 0.5))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ h l)))
        (t_3 (* t_2 d))
        (t_4 (sqrt (/ d l)))
        (t_5 (sqrt (/ d h))))
   (if (<= t_1 (- INFINITY))
     (* (* (/ (fma (/ M d) (* (/ M d) (* (* (* D D) -0.125) h)) l) l) t_4) t_5)
     (if (<= t_1 -5e-89)
       (* (fma t_0 (* t_0 (* (/ h l) -0.5)) 1.0) (sqrt (* (/ (/ d l) h) d)))
       (if (<= t_1 0.0)
         (/ d (sqrt (* l h)))
         (if (<= t_1 5e+253)
           (* t_4 t_5)
           (if (<= t_1 INFINITY)
             (/ t_3 h)
             (/
              (fma (* (* (/ (* D D) d) M) (* -0.125 M)) (* (/ h l) t_2) t_3)
              h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D * M) / d) * 0.5;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h / l));
	double t_3 = t_2 * d;
	double t_4 = sqrt((d / l));
	double t_5 = sqrt((d / h));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((fma((M / d), ((M / d) * (((D * D) * -0.125) * h)), l) / l) * t_4) * t_5;
	} else if (t_1 <= -5e-89) {
		tmp = fma(t_0, (t_0 * ((h / l) * -0.5)), 1.0) * sqrt((((d / l) / h) * d));
	} else if (t_1 <= 0.0) {
		tmp = d / sqrt((l * h));
	} else if (t_1 <= 5e+253) {
		tmp = t_4 * t_5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3 / h;
	} else {
		tmp = fma(((((D * D) / d) * M) * (-0.125 * M)), ((h / l) * t_2), t_3) / h;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h / l))
	t_3 = Float64(t_2 * d)
	t_4 = sqrt(Float64(d / l))
	t_5 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(M / d), Float64(Float64(M / d) * Float64(Float64(Float64(D * D) * -0.125) * h)), l) / l) * t_4) * t_5);
	elseif (t_1 <= -5e-89)
		tmp = Float64(fma(t_0, Float64(t_0 * Float64(Float64(h / l) * -0.5)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (t_1 <= 0.0)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_1 <= 5e+253)
		tmp = Float64(t_4 * t_5);
	elseif (t_1 <= Inf)
		tmp = Float64(t_3 / h);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * D) / d) * M) * Float64(-0.125 * M)), Float64(Float64(h / l) * t_2), t_3) / h);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * d), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$1, -5e-89], N[(N[(t$95$0 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(t$95$4 * t$95$5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$3 / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 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 84.6%

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

   4. Applied rewrites 40.5%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

   9. Applied rewrites 73.0%

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

   11. Applied rewrites 79.6%

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

   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)))) < -4.99999999999999967e-89

   1. Initial program 98.4%

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

   4. Applied rewrites 67.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. clear-num N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

   6. Applied rewrites 67.8%

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

   if -4.99999999999999967e-89 < (*.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 39.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.5%

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

   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)))) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 98.4%

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

   if 4.9999999999999997e253 < (*.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 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.0%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 70.7%

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

   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 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 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 20.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot M}{d} \cdot 0.5, \left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{h}{\ell}}\\ t_4 := t\_3 \cdot d\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_4}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot t\_3, t\_4\right)}{h}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (sqrt (/ h l)))
        (t_4 (* t_3 d)))
   (if (<= t_0 -5e-89)
     (* (* (/ (fma (/ M d) (* (/ M d) (* (* (* D D) -0.125) h)) l) l) t_1) t_2)
     (if (<= t_0 0.0)
       (/ d (sqrt (* l h)))
       (if (<= t_0 5e+253)
         (* t_1 t_2)
         (if (<= t_0 INFINITY)
           (/ t_4 h)
           (/
            (fma (* (* (/ (* D D) d) M) (* -0.125 M)) (* (/ h l) t_3) t_4)
            h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = sqrt((h / l));
	double t_4 = t_3 * d;
	double tmp;
	if (t_0 <= -5e-89) {
		tmp = ((fma((M / d), ((M / d) * (((D * D) * -0.125) * h)), l) / l) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = d / sqrt((l * h));
	} else if (t_0 <= 5e+253) {
		tmp = t_1 * t_2;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_4 / h;
	} else {
		tmp = fma(((((D * D) / d) * M) * (-0.125 * M)), ((h / l) * t_3), t_4) / h;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = sqrt(Float64(h / l))
	t_4 = Float64(t_3 * d)
	tmp = 0.0
	if (t_0 <= -5e-89)
		tmp = Float64(Float64(Float64(fma(Float64(M / d), Float64(Float64(M / d) * Float64(Float64(Float64(D * D) * -0.125) * h)), l) / l) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_0 <= 5e+253)
		tmp = Float64(t_1 * t_2);
	elseif (t_0 <= Inf)
		tmp = Float64(t_4 / h);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * D) / d) * M) * Float64(-0.125 * M)), Float64(Float64(h / l) * t_3), t_4) / h);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * d), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-89], N[(N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+253], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$4 / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -4.99999999999999967e-89

   1. Initial program 87.6%

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 32.1

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 66.5%

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

   11. Applied rewrites 71.7%

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

   if -4.99999999999999967e-89 < (*.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 39.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.5%

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

   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)))) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 98.4%

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

   if 4.9999999999999997e253 < (*.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 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.0%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 70.7%

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

   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 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 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 20.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot D}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := -0.125 \cdot \left(M \cdot M\right)\\ t_4 := \sqrt{\frac{h}{\ell}}\\ t_5 := \frac{h}{\ell} \cdot t\_4\\ t_6 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_7 := t\_4 \cdot d\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(t\_3 \cdot t\_0\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_6 \leq 10^{-264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D \cdot \left(\frac{D}{d} \cdot t\_3\right), t\_5, t\_7\right)}{h}\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{t\_7}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot M\right) \cdot \left(-0.125 \cdot M\right), t\_5, t\_7\right)}{h}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* D D) d))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (* -0.125 (* M M)))
        (t_4 (sqrt (/ h l)))
        (t_5 (* (/ h l) t_4))
        (t_6
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_7 (* t_4 d)))
   (if (<= t_6 (- INFINITY))
     (* (* (* (/ h (* l d)) (* t_3 t_0)) t_1) t_2)
     (if (<= t_6 1e-264)
       (/ (fma (* D (* (/ D d) t_3)) t_5 t_7) h)
       (if (<= t_6 5e+253)
         (* t_1 t_2)
         (if (<= t_6 INFINITY)
           (/ t_7 h)
           (/ (fma (* (* t_0 M) (* -0.125 M)) t_5 t_7) h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * D) / d;
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = -0.125 * (M * M);
	double t_4 = sqrt((h / l));
	double t_5 = (h / l) * t_4;
	double t_6 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_7 = t_4 * d;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (((h / (l * d)) * (t_3 * t_0)) * t_1) * t_2;
	} else if (t_6 <= 1e-264) {
		tmp = fma((D * ((D / d) * t_3)), t_5, t_7) / h;
	} else if (t_6 <= 5e+253) {
		tmp = t_1 * t_2;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_7 / h;
	} else {
		tmp = fma(((t_0 * M) * (-0.125 * M)), t_5, t_7) / h;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * D) / d)
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(-0.125 * Float64(M * M))
	t_4 = sqrt(Float64(h / l))
	t_5 = Float64(Float64(h / l) * t_4)
	t_6 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_7 = Float64(t_4 * d)
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(t_3 * t_0)) * t_1) * t_2);
	elseif (t_6 <= 1e-264)
		tmp = Float64(fma(Float64(D * Float64(Float64(D / d) * t_3)), t_5, t_7) / h);
	elseif (t_6 <= 5e+253)
		tmp = Float64(t_1 * t_2);
	elseif (t_6 <= Inf)
		tmp = Float64(t_7 / h);
	else
		tmp = Float64(fma(Float64(Float64(t_0 * M) * Float64(-0.125 * M)), t_5, t_7) / h);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(h / l), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$7 = N[(t$95$4 * d), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$6, 1e-264], N[(N[(N[(D * N[(N[(D / d), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$5 + t$95$7), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$6, 5e+253], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(t$95$7 / h), $MachinePrecision], N[(N[(N[(N[(t$95$0 * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * t$95$5 + t$95$7), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 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 84.6%

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

   4. Applied rewrites 40.5%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

   9. Applied rewrites 73.0%

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

   10. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-*r/ N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. associate-*r/ N/A

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

   12. Applied rewrites 70.0%

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

   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)))) < 1e-264

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 31.4%

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

   7. Applied rewrites 39.1%

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

   9. Applied rewrites 49.5%

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

   if 1e-264 < (*.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)))) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   7. Applied rewrites 98.4%

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

   if 4.9999999999999997e253 < (*.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 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.0%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 70.7%

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

   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 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 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 20.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_3}{h}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (sqrt (/ h l))))
   (if (<= t_0 -1e-77)
     (* (* (* (/ h (* l d)) (* (* -0.125 (* M M)) (/ (* D D) d))) t_1) t_2)
     (if (<= t_0 0.0)
       (/ d (sqrt (* l h)))
       (if (<= t_0 5e+253)
         (* t_1 t_2)
         (if (<= t_0 INFINITY) (/ (* t_3 d) h) (/ (* (- d) t_3) h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = sqrt((h / l));
	double tmp;
	if (t_0 <= -1e-77) {
		tmp = (((h / (l * d)) * ((-0.125 * (M * M)) * ((D * D) / d))) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = d / sqrt((l * h));
	} else if (t_0 <= 5e+253) {
		tmp = t_1 * t_2;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (t_3 * d) / h;
	} else {
		tmp = (-d * t_3) / h;
	}
	return tmp;
}

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double t_3 = Math.sqrt((h / l));
	double tmp;
	if (t_0 <= -1e-77) {
		tmp = (((h / (l * d)) * ((-0.125 * (M * M)) * ((D * D) / d))) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = d / Math.sqrt((l * h));
	} else if (t_0 <= 5e+253) {
		tmp = t_1 * t_2;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (t_3 * d) / h;
	} else {
		tmp = (-d * t_3) / h;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	t_3 = math.sqrt((h / l))
	tmp = 0
	if t_0 <= -1e-77:
		tmp = (((h / (l * d)) * ((-0.125 * (M * M)) * ((D * D) / d))) * t_1) * t_2
	elif t_0 <= 0.0:
		tmp = d / math.sqrt((l * h))
	elif t_0 <= 5e+253:
		tmp = t_1 * t_2
	elif t_0 <= math.inf:
		tmp = (t_3 * d) / h
	else:
		tmp = (-d * t_3) / h
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = sqrt(Float64(h / l))
	tmp = 0.0
	if (t_0 <= -1e-77)
		tmp = Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(Float64(-0.125 * Float64(M * M)) * Float64(Float64(D * D) / d))) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_0 <= 5e+253)
		tmp = Float64(t_1 * t_2);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(t_3 * d) / h);
	else
		tmp = Float64(Float64(Float64(-d) * t_3) / h);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / l));
	t_2 = sqrt((d / h));
	t_3 = sqrt((h / l));
	tmp = 0.0;
	if (t_0 <= -1e-77)
		tmp = (((h / (l * d)) * ((-0.125 * (M * M)) * ((D * D) / d))) * t_1) * t_2;
	elseif (t_0 <= 0.0)
		tmp = d / sqrt((l * h));
	elseif (t_0 <= 5e+253)
		tmp = t_1 * t_2;
	elseif (t_0 <= Inf)
		tmp = (t_3 * d) / h;
	else
		tmp = (-d * t_3) / h;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-77], N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+253], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(t$95$3 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[((-d) * t$95$3), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -9.9999999999999993e-78

   1. Initial program 87.5%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 32.4

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

   7. Applied rewrites 32.4%

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

   9. Applied rewrites 66.1%

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

   10. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-*r/ N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. associate-*r/ N/A

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

   12. Applied rewrites 61.4%

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

   if -9.9999999999999993e-78 < (*.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 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 d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 54.9%

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

   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)))) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 98.4%

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

   if 4.9999999999999997e253 < (*.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 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.0%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 70.7%

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

   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 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 18.0%

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

   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 17.5%

      \[\leadsto \frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := -0.125 \cdot \left(M \cdot M\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{h}{\ell}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(t\_0 \cdot \frac{D \cdot D}{d}\right)\right) \cdot t\_3\right) \cdot t\_4\\ \mathbf{elif}\;t\_1 \leq 10^{-264} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(D \cdot \left(\frac{D}{d} \cdot t\_0\right), \frac{h}{\ell} \cdot t\_2, t\_2 \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* -0.125 (* M M)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ h l)))
        (t_3 (sqrt (/ d l)))
        (t_4 (sqrt (/ d h))))
   (if (<= t_1 (- INFINITY))
     (* (* (* (/ h (* l d)) (* t_0 (/ (* D D) d))) t_3) t_4)
     (if (or (<= t_1 1e-264) (not (<= t_1 5e+253)))
       (/ (fma (* D (* (/ D d) t_0)) (* (/ h l) t_2) (* t_2 d)) h)
       (* t_3 t_4)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = -0.125 * (M * M);
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h / l));
	double t_3 = sqrt((d / l));
	double t_4 = sqrt((d / h));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((h / (l * d)) * (t_0 * ((D * D) / d))) * t_3) * t_4;
	} else if ((t_1 <= 1e-264) || !(t_1 <= 5e+253)) {
		tmp = fma((D * ((D / d) * t_0)), ((h / l) * t_2), (t_2 * d)) / h;
	} else {
		tmp = t_3 * t_4;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(-0.125 * Float64(M * M))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h / l))
	t_3 = sqrt(Float64(d / l))
	t_4 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(t_0 * Float64(Float64(D * D) / d))) * t_3) * t_4);
	elseif ((t_1 <= 1e-264) || !(t_1 <= 5e+253))
		tmp = Float64(fma(Float64(D * Float64(Float64(D / d) * t_0)), Float64(Float64(h / l) * t_2), Float64(t_2 * d)) / h);
	else
		tmp = Float64(t_3 * t_4);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision], If[Or[LessEqual[t$95$1, 1e-264], N[Not[LessEqual[t$95$1, 5e+253]], $MachinePrecision]], N[(N[(N[(D * N[(N[(D / d), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$3 * t$95$4), $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 84.6%

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

   4. Applied rewrites 40.5%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

   9. Applied rewrites 73.0%

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

   10. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-*r/ N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. associate-*r/ N/A

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

   12. Applied rewrites 70.0%

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

   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)))) < 1e-264 or 4.9999999999999997e253 < (*.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 33.8%

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

   3. Taylor expanded in h around 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 27.0%

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

   7. Applied rewrites 32.8%

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

   9. Applied rewrites 37.0%

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

   if 1e-264 < (*.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)))) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   7. Applied rewrites 98.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-264} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(D \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d} \cdot 0.5\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-71}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \mathsf{fma}\left(\frac{t\_0}{-1} \cdot \frac{t\_0 \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (* D M) d) 0.5)))
   (if (<= d -9.5e-71)
     (*
      (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt (- d)) (sqrt (- l))))
      (fma (* (/ t_0 -1.0) (/ (* t_0 0.5) l)) h 1.0))
     (if (<= d 7.5e-189)
       (/
        (* (sqrt (/ h l)) (fma (/ h l) (/ (* -0.125 (pow (* M D) 2.0)) d) d))
        h)
       (/
        (*
         (fma (* (/ h l) -0.5) (* 0.25 (pow (/ d (* D M)) -2.0)) 1.0)
         (/ d (sqrt l)))
        (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D * M) / d) * 0.5;
	double tmp;
	if (d <= -9.5e-71) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) / sqrt(-l))) * fma(((t_0 / -1.0) * ((t_0 * 0.5) / l)), h, 1.0);
	} else if (d <= 7.5e-189) {
		tmp = (sqrt((h / l)) * fma((h / l), ((-0.125 * pow((M * D), 2.0)) / d), d)) / h;
	} else {
		tmp = (fma(((h / l) * -0.5), (0.25 * pow((d / (D * M)), -2.0)), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D * M) / d) * 0.5)
	tmp = 0.0
	if (d <= -9.5e-71)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * fma(Float64(Float64(t_0 / -1.0) * Float64(Float64(t_0 * 0.5) / l)), h, 1.0));
	elseif (d <= 7.5e-189)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(h / l), Float64(Float64(-0.125 * (Float64(M * D) ^ 2.0)) / d), d)) / h);
	else
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), Float64(0.25 * (Float64(d / Float64(D * M)) ^ -2.0)), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[d, -9.5e-71], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / -1.0), $MachinePrecision] * N[(N[(t$95$0 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e-189], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.25 * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.4999999999999994e-71

   1. Initial program 74.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 82.7

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

   6. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 82.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if -9.4999999999999994e-71 < d < 7.50000000000000042e-189

   1. Initial program 40.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.1%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 73.6%

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

   if 7.50000000000000042e-189 < d

   1. Initial program 74.8%

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

   4. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 86.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-71}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d} \cdot 0.5\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{t\_0}{-1} \cdot \frac{t\_0 \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (* D M) d) 0.5)))
   (if (<= d -2.1e+107)
     (*
      (* (- d) (sqrt (pow (* l h) -1.0)))
      (fma (/ (* -0.5 (pow (* (/ d M) (/ 2.0 D)) -2.0)) l) h 1.0))
     (if (<= d -1.02e-47)
       (*
        (* (pow (sqrt (/ h d)) -1.0) (sqrt (/ d l)))
        (fma (* (/ t_0 -1.0) (/ (* t_0 0.5) l)) h 1.0))
       (if (<= d 7.5e-189)
         (/
          (* (sqrt (/ h l)) (fma (/ h l) (/ (* -0.125 (pow (* M D) 2.0)) d) d))
          h)
         (/
          (*
           (fma (* (/ h l) -0.5) (* 0.25 (pow (/ d (* D M)) -2.0)) 1.0)
           (/ d (sqrt l)))
          (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D * M) / d) * 0.5;
	double tmp;
	if (d <= -2.1e+107) {
		tmp = (-d * sqrt(pow((l * h), -1.0))) * fma(((-0.5 * pow(((d / M) * (2.0 / D)), -2.0)) / l), h, 1.0);
	} else if (d <= -1.02e-47) {
		tmp = (pow(sqrt((h / d)), -1.0) * sqrt((d / l))) * fma(((t_0 / -1.0) * ((t_0 * 0.5) / l)), h, 1.0);
	} else if (d <= 7.5e-189) {
		tmp = (sqrt((h / l)) * fma((h / l), ((-0.125 * pow((M * D), 2.0)) / d), d)) / h;
	} else {
		tmp = (fma(((h / l) * -0.5), (0.25 * pow((d / (D * M)), -2.0)), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D * M) / d) * 0.5)
	tmp = 0.0
	if (d <= -2.1e+107)
		tmp = Float64(Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0))) * fma(Float64(Float64(-0.5 * (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0)) / l), h, 1.0));
	elseif (d <= -1.02e-47)
		tmp = Float64(Float64((sqrt(Float64(h / d)) ^ -1.0) * sqrt(Float64(d / l))) * fma(Float64(Float64(t_0 / -1.0) * Float64(Float64(t_0 * 0.5) / l)), h, 1.0));
	elseif (d <= 7.5e-189)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(h / l), Float64(Float64(-0.125 * (Float64(M * D) ^ 2.0)) / d), d)) / h);
	else
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), Float64(0.25 * (Float64(d / Float64(D * M)) ^ -2.0)), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[d, -2.1e+107], N[(N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.02e-47], N[(N[(N[Power[N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / -1.0), $MachinePrecision] * N[(N[(t$95$0 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e-189], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.25 * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.1e107

   1. Initial program 67.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 77.9

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

   6. Applied rewrites 77.9%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.2

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

   9. Applied rewrites 92.2%

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

   if -2.1e107 < d < -1.02000000000000002e-47

   1. Initial program 77.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 85.4

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

   6. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 87.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.9

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

   10. Applied rewrites 87.9%

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

   if -1.02000000000000002e-47 < d < 7.50000000000000042e-189

   1. Initial program 44.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.1%

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

   7. Applied rewrites 47.6%

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

   9. Applied rewrites 75.3%

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

   if 7.50000000000000042e-189 < d

   1. Initial program 74.8%

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

   4. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 86.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \frac{D \cdot M}{d} \cdot 0.5\\ t_2 := \frac{M \cdot D}{d}\\ t_3 := \left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{t\_1}{-1} \cdot \frac{t\_1 \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot t\_2\right), h \cdot \frac{0.25 \cdot t\_2}{\ell}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1 (* (/ (* D M) d) 0.5))
        (t_2 (/ (* M D) d))
        (t_3
         (*
          (* (pow (sqrt (/ h d)) -1.0) (sqrt (/ d l)))
          (fma (* (/ t_1 -1.0) (/ (* t_1 0.5) l)) h 1.0))))
   (if (<= d -1.02e-47)
     t_3
     (if (<= d 1.1e-272)
       (/
        (* (sqrt (/ h l)) (fma (/ h l) (/ (* -0.125 (pow (* M D) 2.0)) d) d))
        h)
       (if (<= d 6.3e-86)
         (fma (* t_0 (* -0.5 t_2)) (* h (/ (* 0.25 t_2) l)) t_0)
         t_3)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = ((D * M) / d) * 0.5;
	double t_2 = (M * D) / d;
	double t_3 = (pow(sqrt((h / d)), -1.0) * sqrt((d / l))) * fma(((t_1 / -1.0) * ((t_1 * 0.5) / l)), h, 1.0);
	double tmp;
	if (d <= -1.02e-47) {
		tmp = t_3;
	} else if (d <= 1.1e-272) {
		tmp = (sqrt((h / l)) * fma((h / l), ((-0.125 * pow((M * D), 2.0)) / d), d)) / h;
	} else if (d <= 6.3e-86) {
		tmp = fma((t_0 * (-0.5 * t_2)), (h * ((0.25 * t_2) / l)), t_0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_2 = Float64(Float64(M * D) / d)
	t_3 = Float64(Float64((sqrt(Float64(h / d)) ^ -1.0) * sqrt(Float64(d / l))) * fma(Float64(Float64(t_1 / -1.0) * Float64(Float64(t_1 * 0.5) / l)), h, 1.0))
	tmp = 0.0
	if (d <= -1.02e-47)
		tmp = t_3;
	elseif (d <= 1.1e-272)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(h / l), Float64(Float64(-0.125 * (Float64(M * D) ^ 2.0)) / d), d)) / h);
	elseif (d <= 6.3e-86)
		tmp = fma(Float64(t_0 * Float64(-0.5 * t_2)), Float64(h * Float64(Float64(0.25 * t_2) / l)), t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 / -1.0), $MachinePrecision] * N[(N[(t$95$1 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.02e-47], t$95$3, If[LessEqual[d, 1.1e-272], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 6.3e-86], N[(N[(t$95$0 * N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.02000000000000002e-47 or 6.2999999999999999e-86 < d

   1. Initial program 77.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 84.1

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

   6. Applied rewrites 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.2

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 85.5

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

   10. Applied rewrites 85.5%

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

   if -1.02000000000000002e-47 < d < 1.09999999999999994e-272

   1. Initial program 46.8%

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

   3. Taylor expanded in h around 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 44.7%

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

   7. Applied rewrites 50.7%

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

   9. Applied rewrites 75.7%

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

   if 1.09999999999999994e-272 < d < 6.2999999999999999e-86

   1. Initial program 37.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 37.6

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

   6. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 76.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \frac{D \cdot M}{d} \cdot 0.5\\ t_2 := \mathsf{fma}\left(\frac{t\_1}{-1} \cdot \frac{t\_1 \cdot 0.5}{\ell}, h, 1\right)\\ t_3 := \frac{M \cdot D}{d}\\ t_4 := \sqrt{\frac{h}{\ell}}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-178}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t\_5\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot t\_4, t\_4 \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot t\_3\right), h \cdot \frac{0.25 \cdot t\_3}{\ell}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot t\_5\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1 (* (/ (* D M) d) 0.5))
        (t_2 (fma (* (/ t_1 -1.0) (/ (* t_1 0.5) l)) h 1.0))
        (t_3 (/ (* M D) d))
        (t_4 (sqrt (/ h l)))
        (t_5 (sqrt (/ d l))))
   (if (<= d -2.6e-178)
     (* (* (sqrt (/ d h)) t_5) t_2)
     (if (<= d 2.3e-273)
       (/
        (fma (* (* (/ (* D D) d) M) (* -0.125 M)) (* (/ h l) t_4) (* t_4 d))
        h)
       (if (<= d 6.3e-86)
         (fma (* t_0 (* -0.5 t_3)) (* h (/ (* 0.25 t_3) l)) t_0)
         (* (* (pow (sqrt (/ h d)) -1.0) t_5) t_2))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = ((D * M) / d) * 0.5;
	double t_2 = fma(((t_1 / -1.0) * ((t_1 * 0.5) / l)), h, 1.0);
	double t_3 = (M * D) / d;
	double t_4 = sqrt((h / l));
	double t_5 = sqrt((d / l));
	double tmp;
	if (d <= -2.6e-178) {
		tmp = (sqrt((d / h)) * t_5) * t_2;
	} else if (d <= 2.3e-273) {
		tmp = fma(((((D * D) / d) * M) * (-0.125 * M)), ((h / l) * t_4), (t_4 * d)) / h;
	} else if (d <= 6.3e-86) {
		tmp = fma((t_0 * (-0.5 * t_3)), (h * ((0.25 * t_3) / l)), t_0);
	} else {
		tmp = (pow(sqrt((h / d)), -1.0) * t_5) * t_2;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_2 = fma(Float64(Float64(t_1 / -1.0) * Float64(Float64(t_1 * 0.5) / l)), h, 1.0)
	t_3 = Float64(Float64(M * D) / d)
	t_4 = sqrt(Float64(h / l))
	t_5 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -2.6e-178)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_5) * t_2);
	elseif (d <= 2.3e-273)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * D) / d) * M) * Float64(-0.125 * M)), Float64(Float64(h / l) * t_4), Float64(t_4 * d)) / h);
	elseif (d <= 6.3e-86)
		tmp = fma(Float64(t_0 * Float64(-0.5 * t_3)), Float64(h * Float64(Float64(0.25 * t_3) / l)), t_0);
	else
		tmp = Float64(Float64((sqrt(Float64(h / d)) ^ -1.0) * t_5) * t_2);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 / -1.0), $MachinePrecision] * N[(N[(t$95$1 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.6e-178], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 2.3e-273], N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$4 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 6.3e-86], N[(N[(t$95$0 * N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$3), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[Power[N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.59999999999999998e-178

   1. Initial program 71.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 79.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 79.3

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

   10. Applied rewrites 79.3%

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

   if -2.59999999999999998e-178 < d < 2.29999999999999981e-273

   1. Initial program 36.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.7%

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

   7. Applied rewrites 51.1%

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

   9. Applied rewrites 53.9%

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

   if 2.29999999999999981e-273 < d < 6.2999999999999999e-86

   1. Initial program 37.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 37.6

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

   6. Applied rewrites 37.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 76.9%

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

   if 6.2999999999999999e-86 < d

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.6

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

   6. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.7

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 89.1

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

   10. Applied rewrites 89.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-178}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\frac{h}{d}}\right)}^{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_2 := \left(D \cdot \frac{0.5}{d}\right) \cdot M\\ t_3 := \frac{M \cdot D}{d}\\ t_4 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -7.9 \cdot 10^{-109}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot t\_4, t\_4 \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(-0.5 \cdot t\_3\right), h \cdot \frac{0.25 \cdot t\_3}{\ell}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot t\_2, t\_2, 1\right) \cdot t\_0\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ d (sqrt (* h l))))
        (t_2 (* (* D (/ 0.5 d)) M))
        (t_3 (/ (* M D) d))
        (t_4 (sqrt (/ h l))))
   (if (<= d -3.6e+103)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -7.9e-109)
       (*
        (* (/ (fma (/ M d) (* (/ M d) (* (* (* D D) -0.125) h)) l) l) t_0)
        (sqrt (/ d h)))
       (if (<= d 2.3e-273)
         (/
          (fma (* (* (/ (* D D) d) M) (* -0.125 M)) (* (/ h l) t_4) (* t_4 d))
          h)
         (if (<= d 7.2e-107)
           (fma (* t_1 (* -0.5 t_3)) (* h (/ (* 0.25 t_3) l)) t_1)
           (/
            (* (* (fma (* (* (/ h l) -0.5) t_2) t_2 1.0) t_0) (sqrt d))
            (sqrt h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = d / sqrt((h * l));
	double t_2 = (D * (0.5 / d)) * M;
	double t_3 = (M * D) / d;
	double t_4 = sqrt((h / l));
	double tmp;
	if (d <= -3.6e+103) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -7.9e-109) {
		tmp = ((fma((M / d), ((M / d) * (((D * D) * -0.125) * h)), l) / l) * t_0) * sqrt((d / h));
	} else if (d <= 2.3e-273) {
		tmp = fma(((((D * D) / d) * M) * (-0.125 * M)), ((h / l) * t_4), (t_4 * d)) / h;
	} else if (d <= 7.2e-107) {
		tmp = fma((t_1 * (-0.5 * t_3)), (h * ((0.25 * t_3) / l)), t_1);
	} else {
		tmp = ((fma((((h / l) * -0.5) * t_2), t_2, 1.0) * t_0) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	t_2 = Float64(Float64(D * Float64(0.5 / d)) * M)
	t_3 = Float64(Float64(M * D) / d)
	t_4 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -3.6e+103)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -7.9e-109)
		tmp = Float64(Float64(Float64(fma(Float64(M / d), Float64(Float64(M / d) * Float64(Float64(Float64(D * D) * -0.125) * h)), l) / l) * t_0) * sqrt(Float64(d / h)));
	elseif (d <= 2.3e-273)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * D) / d) * M) * Float64(-0.125 * M)), Float64(Float64(h / l) * t_4), Float64(t_4 * d)) / h);
	elseif (d <= 7.2e-107)
		tmp = fma(Float64(t_1 * Float64(-0.5 * t_3)), Float64(h * Float64(Float64(0.25 * t_3) / l)), t_1);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * t_2), t_2, 1.0) * t_0) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.6e+103], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.9e-109], N[(N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e-273], N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$4 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7.2e-107], N[(N[(t$95$1 * N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$3), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.60000000000000017e103

   1. Initial program 69.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -3.60000000000000017e103 < d < -7.8999999999999997e-109

   1. Initial program 77.2%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   9. Applied rewrites 69.3%

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

   11. Applied rewrites 69.7%

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

   if -7.8999999999999997e-109 < d < 2.29999999999999981e-273

   1. Initial program 39.5%

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

   3. Taylor expanded in 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 49.1%

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

   7. Applied rewrites 51.1%

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

   9. Applied rewrites 53.4%

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

   if 2.29999999999999981e-273 < d < 7.19999999999999953e-107

   1. Initial program 34.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 34.9

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

   6. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

   8. Applied rewrites 35.0%

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

   10. Applied rewrites 79.2%

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

   if 7.19999999999999953e-107 < d

   1. Initial program 80.3%

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

   4. Applied rewrites 86.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. pow-pow N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-1}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. inv-pow N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{d}{M} \cdot \frac{2}{D}}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d}{M} \cdot \frac{2}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d}{M}} \cdot \frac{2}{D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{d}{M} \cdot \color{blue}{\frac{2}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. frac-times N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d \cdot 2}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. clear-num N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   6. Applied rewrites 87.2%

      \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right), \left(D \cdot \frac{0.5}{d}\right) \cdot M, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -7.9 \cdot 10^{-109}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right), \left(D \cdot \frac{0.5}{d}\right) \cdot M, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \left(D \cdot \frac{0.5}{d}\right) \cdot M\\ t_2 := \frac{D \cdot M}{d} \cdot 0.5\\ t_3 := \frac{M \cdot D}{d}\\ t_4 := \sqrt{\frac{h}{\ell}}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-178}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t\_5\right) \cdot \mathsf{fma}\left(\frac{t\_2}{-1} \cdot \frac{t\_2 \cdot 0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot t\_4, t\_4 \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot t\_3\right), h \cdot \frac{0.25 \cdot t\_3}{\ell}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot t\_1, t\_1, 1\right) \cdot t\_5\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1 (* (* D (/ 0.5 d)) M))
        (t_2 (* (/ (* D M) d) 0.5))
        (t_3 (/ (* M D) d))
        (t_4 (sqrt (/ h l)))
        (t_5 (sqrt (/ d l))))
   (if (<= d -2.6e-178)
     (* (* (sqrt (/ d h)) t_5) (fma (* (/ t_2 -1.0) (/ (* t_2 0.5) l)) h 1.0))
     (if (<= d 2.3e-273)
       (/
        (fma (* (* (/ (* D D) d) M) (* -0.125 M)) (* (/ h l) t_4) (* t_4 d))
        h)
       (if (<= d 7.2e-107)
         (fma (* t_0 (* -0.5 t_3)) (* h (/ (* 0.25 t_3) l)) t_0)
         (/
          (* (* (fma (* (* (/ h l) -0.5) t_1) t_1 1.0) t_5) (sqrt d))
          (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = (D * (0.5 / d)) * M;
	double t_2 = ((D * M) / d) * 0.5;
	double t_3 = (M * D) / d;
	double t_4 = sqrt((h / l));
	double t_5 = sqrt((d / l));
	double tmp;
	if (d <= -2.6e-178) {
		tmp = (sqrt((d / h)) * t_5) * fma(((t_2 / -1.0) * ((t_2 * 0.5) / l)), h, 1.0);
	} else if (d <= 2.3e-273) {
		tmp = fma(((((D * D) / d) * M) * (-0.125 * M)), ((h / l) * t_4), (t_4 * d)) / h;
	} else if (d <= 7.2e-107) {
		tmp = fma((t_0 * (-0.5 * t_3)), (h * ((0.25 * t_3) / l)), t_0);
	} else {
		tmp = ((fma((((h / l) * -0.5) * t_1), t_1, 1.0) * t_5) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = Float64(Float64(D * Float64(0.5 / d)) * M)
	t_2 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_3 = Float64(Float64(M * D) / d)
	t_4 = sqrt(Float64(h / l))
	t_5 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -2.6e-178)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_5) * fma(Float64(Float64(t_2 / -1.0) * Float64(Float64(t_2 * 0.5) / l)), h, 1.0));
	elseif (d <= 2.3e-273)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * D) / d) * M) * Float64(-0.125 * M)), Float64(Float64(h / l) * t_4), Float64(t_4 * d)) / h);
	elseif (d <= 7.2e-107)
		tmp = fma(Float64(t_0 * Float64(-0.5 * t_3)), Float64(h * Float64(Float64(0.25 * t_3) / l)), t_0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * t_1), t_1, 1.0) * t_5) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.6e-178], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(N[(t$95$2 / -1.0), $MachinePrecision] * N[(N[(t$95$2 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e-273], N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$4 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7.2e-107], N[(N[(t$95$0 * N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$3), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * t$95$5), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.59999999999999998e-178

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\frac{\ell}{h}}} + 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\color{blue}{\ell \cdot \frac{1}{h}}} + 1\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \frac{1}{\frac{1}{h}}} + 1\right) \]

      11. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{\frac{h}{1}} + 1\right) \]

      12. /-rgt-identity N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{h} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 78.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. lift-sqrt.f64 78.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

   6. Applied rewrites 78.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}}{\ell}, h, 1\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{-1}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{-1} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\frac{-2}{2}}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\frac{-2}{2} \cdot \ell}}, h, 1\right) \]

   8. Applied rewrites 79.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}}, h, 1\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot \frac{1}{2}}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}{\ell}, h, 1\right) \]

      2. metadata-eval 79.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot \frac{1}{2}}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}{\ell}, h, 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot \frac{1}{2}}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}{\ell}, h, 1\right) \]

      5. lift-sqrt.f64 79.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right) \]

   10. Applied rewrites 79.3%

       \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}, h, 1\right) \]

   if -2.59999999999999998e-178 < d < 2.29999999999999981e-273

   1. Initial program 36.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot \left(-0.125 \cdot M\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 2.29999999999999981e-273 < d < 7.19999999999999953e-107

   1. Initial program 34.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\frac{\ell}{h}}} + 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\color{blue}{\ell \cdot \frac{1}{h}}} + 1\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \frac{1}{\frac{1}{h}}} + 1\right) \]

      11. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{\frac{h}{1}} + 1\right) \]

      12. /-rgt-identity N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{h} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. lift-sqrt.f64 34.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

   6. Applied rewrites 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}}{\ell}, h, 1\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{-1}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{-1} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\frac{-2}{2}}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\frac{-2}{2} \cdot \ell}}, h, 1\right) \]

   8. Applied rewrites 35.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}}, h, 1\right) \]

   10. Applied rewrites 79.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   if 7.19999999999999953e-107 < d

   1. Initial program 80.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. pow-pow N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-1}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. inv-pow N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{d}{M} \cdot \frac{2}{D}}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d}{M} \cdot \frac{2}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d}{M}} \cdot \frac{2}{D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{d}{M} \cdot \color{blue}{\frac{2}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. frac-times N/A

         \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{d \cdot 2}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. clear-num N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   6. Applied rewrites 87.2%

      \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right), \left(D \cdot \frac{0.5}{d}\right) \cdot M, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ t_3 := \frac{D \cdot M}{d} \cdot 0.5\\ t_4 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;h \leq -190000000:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot t\_1\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -7.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \frac{D}{d}, \frac{h}{\ell} \cdot t\_2, t\_2 \cdot d\right)}{h}\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, t\_3 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;h \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t\_4 \cdot \left(-0.5 \cdot t\_0\right), h \cdot \frac{0.25 \cdot t\_0}{\ell}, t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot M}{d} \cdot \frac{M}{d}, \ell\right)}{\ell} \cdot t\_1\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ h l)))
        (t_3 (* (/ (* D M) d) 0.5))
        (t_4 (/ d (sqrt (* h l)))))
   (if (<= h -190000000.0)
     (*
      (* (/ (fma (/ M d) (* (/ M d) (* (* (* D D) -0.125) h)) l) l) t_1)
      (sqrt (/ d h)))
     (if (<= h -7.8e-104)
       (/
        (fma (* (* (* -0.125 (* M M)) D) (/ D d)) (* (/ h l) t_2) (* t_2 d))
        h)
       (if (<= h 5.5e-102)
         (* (fma t_3 (* t_3 (* (/ h l) -0.5)) 1.0) (sqrt (* (/ (/ d l) h) d)))
         (if (<= h 1.4e+138)
           (fma (* t_4 (* -0.5 t_0)) (* h (/ (* 0.25 t_0) l)) t_4)
           (/
            (*
             (* (/ (fma (* -0.125 (* D D)) (* (/ (* h M) d) (/ M d)) l) l) t_1)
             (sqrt d))
            (sqrt h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / l));
	double t_3 = ((D * M) / d) * 0.5;
	double t_4 = d / sqrt((h * l));
	double tmp;
	if (h <= -190000000.0) {
		tmp = ((fma((M / d), ((M / d) * (((D * D) * -0.125) * h)), l) / l) * t_1) * sqrt((d / h));
	} else if (h <= -7.8e-104) {
		tmp = fma((((-0.125 * (M * M)) * D) * (D / d)), ((h / l) * t_2), (t_2 * d)) / h;
	} else if (h <= 5.5e-102) {
		tmp = fma(t_3, (t_3 * ((h / l) * -0.5)), 1.0) * sqrt((((d / l) / h) * d));
	} else if (h <= 1.4e+138) {
		tmp = fma((t_4 * (-0.5 * t_0)), (h * ((0.25 * t_0) / l)), t_4);
	} else {
		tmp = (((fma((-0.125 * (D * D)), (((h * M) / d) * (M / d)), l) / l) * t_1) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / l))
	t_3 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_4 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (h <= -190000000.0)
		tmp = Float64(Float64(Float64(fma(Float64(M / d), Float64(Float64(M / d) * Float64(Float64(Float64(D * D) * -0.125) * h)), l) / l) * t_1) * sqrt(Float64(d / h)));
	elseif (h <= -7.8e-104)
		tmp = Float64(fma(Float64(Float64(Float64(-0.125 * Float64(M * M)) * D) * Float64(D / d)), Float64(Float64(h / l) * t_2), Float64(t_2 * d)) / h);
	elseif (h <= 5.5e-102)
		tmp = Float64(fma(t_3, Float64(t_3 * Float64(Float64(h / l) * -0.5)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (h <= 1.4e+138)
		tmp = fma(Float64(t_4 * Float64(-0.5 * t_0)), Float64(h * Float64(Float64(0.25 * t_0) / l)), t_4);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-0.125 * Float64(D * D)), Float64(Float64(Float64(h * M) / d) * Float64(M / d)), l) / l) * t_1) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $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[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -190000000.0], N[(N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -7.8e-104], N[(N[(N[(N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[h, 5.5e-102], N[(N[(t$95$3 * N[(t$95$3 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.4e+138], N[(N[(t$95$4 * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.25 * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -1.9e8

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \ell}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot {D}^{2}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {D}^{2}}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. associate-/r* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      16. lower-*.f64 0.0

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   7. Applied rewrites 0.0%

      \[\leadsto \frac{\left(\color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{M \cdot M}{d}}{d} \cdot h, \ell\right)}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   9. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(\frac{d}{M}\right)}^{-2} \cdot h, \left(D \cdot D\right) \cdot -0.125, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 57.1%

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -1.9e8 < h < -7.8000000000000004e-104

   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. 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 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \frac{D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if -7.8000000000000004e-104 < h < 5.4999999999999997e-102

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\left(\color{blue}{\frac{-2}{2}} \cdot 2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      6. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\left(\frac{-2}{2}\right)}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      7. metadata-eval N/A

         \[\leadsto \left({\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{-1}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      8. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{d}{M} \cdot \frac{2}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d}{M} \cdot \frac{2}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d}{M}} \cdot \frac{2}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{1}{\frac{d}{M} \cdot \color{blue}{\frac{2}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      12. frac-times N/A

          \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d \cdot 2}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      14. clear-num N/A

          \[\leadsto \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      15. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      16. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   6. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{D \cdot M}{d} \cdot 0.5, \left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   if 5.4999999999999997e-102 < h < 1.4e138

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\frac{\ell}{h}}} + 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot 1}{\color{blue}{\ell \cdot \frac{1}{h}}} + 1\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \frac{1}{\frac{1}{h}}} + 1\right) \]

      11. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{\frac{h}{1}} + 1\right) \]

      12. /-rgt-identity N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot \color{blue}{h} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 73.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. lift-sqrt.f64 73.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

   6. Applied rewrites 73.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}}{\ell}, h, 1\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}}, h, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{-1}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{-1} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\frac{-2}{2}}} \cdot \frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, h, 1\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\frac{-2}{2} \cdot \ell}}, h, 1\right) \]

   8. Applied rewrites 73.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{D \cdot M}{d} \cdot 0.5}{-1} \cdot \frac{\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot 0.5}{\ell}}, h, 1\right) \]

   10. Applied rewrites 88.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.5 \cdot \frac{M \cdot D}{d}\right), h \cdot \frac{0.25 \cdot \frac{M \cdot D}{d}}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   if 1.4e138 < h

   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

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \ell}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot {D}^{2}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {D}^{2}}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. associate-/r* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      16. lower-*.f64 68.6

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   7. Applied rewrites 68.6%

      \[\leadsto \frac{\left(\color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{M \cdot M}{d}}{d} \cdot h, \ell\right)}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.9%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot M}{d} \cdot \frac{M}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d} \cdot 0.5\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;h \leq -190000000:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot t\_1\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -7.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \frac{D}{d}, \frac{h}{\ell} \cdot t\_2, t\_2 \cdot d\right)}{h}\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot M}{d} \cdot \frac{M}{d}, \ell\right)}{\ell} \cdot t\_1\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (* D M) d) 0.5)) (t_1 (sqrt (/ d l))) (t_2 (sqrt (/ h l))))
   (if (<= h -190000000.0)
     (*
      (* (/ (fma (/ M d) (* (/ M d) (* (* (* D D) -0.125) h)) l) l) t_1)
      (sqrt (/ d h)))
     (if (<= h -7.8e-104)
       (/
        (fma (* (* (* -0.125 (* M M)) D) (/ D d)) (* (/ h l) t_2) (* t_2 d))
        h)
       (if (<= h 3.5e+15)
         (* (fma t_0 (* t_0 (* (/ h l) -0.5)) 1.0) (sqrt (* (/ (/ d l) h) d)))
         (/
          (*
           (* (/ (fma (* -0.125 (* D D)) (* (/ (* h M) d) (/ M d)) l) l) t_1)
           (sqrt d))
          (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D * M) / d) * 0.5;
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / l));
	double tmp;
	if (h <= -190000000.0) {
		tmp = ((fma((M / d), ((M / d) * (((D * D) * -0.125) * h)), l) / l) * t_1) * sqrt((d / h));
	} else if (h <= -7.8e-104) {
		tmp = fma((((-0.125 * (M * M)) * D) * (D / d)), ((h / l) * t_2), (t_2 * d)) / h;
	} else if (h <= 3.5e+15) {
		tmp = fma(t_0, (t_0 * ((h / l) * -0.5)), 1.0) * sqrt((((d / l) / h) * d));
	} else {
		tmp = (((fma((-0.125 * (D * D)), (((h * M) / d) * (M / d)), l) / l) * t_1) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D * M) / d) * 0.5)
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (h <= -190000000.0)
		tmp = Float64(Float64(Float64(fma(Float64(M / d), Float64(Float64(M / d) * Float64(Float64(Float64(D * D) * -0.125) * h)), l) / l) * t_1) * sqrt(Float64(d / h)));
	elseif (h <= -7.8e-104)
		tmp = Float64(fma(Float64(Float64(Float64(-0.125 * Float64(M * M)) * D) * Float64(D / d)), Float64(Float64(h / l) * t_2), Float64(t_2 * d)) / h);
	elseif (h <= 3.5e+15)
		tmp = Float64(fma(t_0, Float64(t_0 * Float64(Float64(h / l) * -0.5)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-0.125 * Float64(D * D)), Float64(Float64(Float64(h * M) / d) * Float64(M / d)), l) / l) * t_1) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.5), $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[h, -190000000.0], N[(N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -7.8e-104], N[(N[(N[(N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[h, 3.5e+15], N[(N[(t$95$0 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.9e8

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \ell}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot {D}^{2}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {D}^{2}}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. associate-/r* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      16. lower-*.f64 0.0

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   7. Applied rewrites 0.0%

      \[\leadsto \frac{\left(\color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{M \cdot M}{d}}{d} \cdot h, \ell\right)}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   9. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(\frac{d}{M}\right)}^{-2} \cdot h, \left(D \cdot D\right) \cdot -0.125, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 57.1%

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{M}{d}, \frac{M}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot h\right), \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -1.9e8 < h < -7.8000000000000004e-104

   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. 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 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \frac{D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if -7.8000000000000004e-104 < h < 3.5e15

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\left(\color{blue}{\frac{-2}{2}} \cdot 2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      6. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\left(\frac{-2}{2}\right)}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      7. metadata-eval N/A

         \[\leadsto \left({\left({\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{\color{blue}{-1}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      8. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{d}{M} \cdot \frac{2}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d}{M} \cdot \frac{2}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d}{M}} \cdot \frac{2}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{1}{\frac{d}{M} \cdot \color{blue}{\frac{2}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      12. frac-times N/A

          \[\leadsto \left({\left(\frac{1}{\color{blue}{\frac{d \cdot 2}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      14. clear-num N/A

          \[\leadsto \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      15. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      16. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   6. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{D \cdot M}{d} \cdot 0.5, \left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   if 3.5e15 < h

   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

   4. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \ell}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} + \ell}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot {D}^{2}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {D}^{2}}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}, \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{{M}^{2}}{{d}^{2}} \cdot h}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      12. associate-/r* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

      16. lower-*.f64 64.0

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\left(\color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{M \cdot M}{d}}{d} \cdot h, \ell\right)}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(D \cdot D\right), \frac{{M}^{2} \cdot h}{{d}^{2}}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.0%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot M}{d} \cdot \frac{M}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.26 \cdot 10^{+51}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{\frac{-d}{\ell} \cdot d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.26e+51)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d -3.2e-167)
     (/ (sqrt (* (/ (- d) l) d)) (sqrt (- h)))
     (if (<= d -1e-310)
       (* (* (* 0.125 (* D D)) (/ (* M M) d)) (/ (sqrt (/ h l)) (fabs l)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.26e+51) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -3.2e-167) {
		tmp = sqrt(((-d / l) * d)) / sqrt(-h);
	} else if (d <= -1e-310) {
		tmp = ((0.125 * (D * D)) * ((M * M) / d)) * (sqrt((h / l)) / fabs(l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.26d+51)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (d <= (-3.2d-167)) then
        tmp = sqrt(((-d / l) * d)) / sqrt(-h)
    else if (d <= (-1d-310)) then
        tmp = ((0.125d0 * (d_1 * d_1)) * ((m * m) / d)) * (sqrt((h / l)) / abs(l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.26e+51) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (d <= -3.2e-167) {
		tmp = Math.sqrt(((-d / l) * d)) / Math.sqrt(-h);
	} else if (d <= -1e-310) {
		tmp = ((0.125 * (D * D)) * ((M * M) / d)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.26e+51:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif d <= -3.2e-167:
		tmp = math.sqrt(((-d / l) * d)) / math.sqrt(-h)
	elif d <= -1e-310:
		tmp = ((0.125 * (D * D)) * ((M * M) / d)) * (math.sqrt((h / l)) / math.fabs(l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.26e+51)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -3.2e-167)
		tmp = Float64(sqrt(Float64(Float64(Float64(-d) / l) * d)) / sqrt(Float64(-h)));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * M) / d)) * Float64(sqrt(Float64(h / l)) / abs(l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.26e+51)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (d <= -3.2e-167)
		tmp = sqrt(((-d / l) * d)) / sqrt(-h);
	elseif (d <= -1e-310)
		tmp = ((0.125 * (D * D)) * ((M * M) / d)) * (sqrt((h / l)) / abs(l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.26e+51], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-167], N[(N[Sqrt[N[(N[((-d) / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.25999999999999997e51

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 63.2

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.25999999999999997e51 < d < -3.2000000000000002e-167

   1. Initial program 81.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 10.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 10.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.0%

      \[\leadsto \frac{\sqrt{\frac{-d}{\ell} \cdot d}}{\color{blue}{\sqrt{-h}}} \]

   if -3.2000000000000002e-167 < d < -9.999999999999969e-311

   1. Initial program 36.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

   5. Applied rewrites 33.6%

      \[\leadsto \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.3%

      \[\leadsto \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\left|\ell\right|}} \]

   if -9.999999999999969e-311 < d

   1. Initial program 65.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 40.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.8%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.26 \cdot 10^{+51}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{\frac{-d}{\ell} \cdot d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-284}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.2e-284)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.2e-284) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.2d-284) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.2e-284) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.2e-284:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.2e-284)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.2e-284)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.2e-284], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2000000000000001e-284

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 47.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.2000000000000001e-284 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.7%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-284}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-284}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.35e-284)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (sqrt (* l h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.35e-284) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.35d-284) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.35e-284) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.35e-284:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.35e-284)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.35e-284)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.35e-284], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.35000000000000011e-284

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 47.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.35000000000000011e-284 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-284}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 24.9

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 24.9%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 25.1%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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)))))