Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 78.2%

Time: 17.9s

Alternatives: 22

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\\ t_2 := \frac{M}{d} \cdot D\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot t\_1, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{t\_0}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-175}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot t\_1\right) \cdot \left(\left(\sqrt{{\left(-\ell\right)}^{-1}} \cdot t\_0\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t\_2 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_2, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (/ (* (* D (/ 0.5 d)) M) l))
        (t_2 (* (/ M d) D)))
   (if (<= d -1.3e+118)
     (*
      (fma (* (* (* (- D) 0.25) (/ M d)) t_1) h 1.0)
      (* (pow (/ d l) (/ 1.0 2.0)) (/ t_0 (sqrt (- h)))))
     (if (<= d -6e-175)
       (*
        (- 1.0 (* (/ (* (* (* D 0.5) 0.5) (/ M d)) (pow h -1.0)) t_1))
        (* (* (sqrt (pow (- l) -1.0)) t_0) (pow (/ d h) (/ 1.0 2.0))))
       (if (<= d 2.4e-182)
         (/
          (fma
           (pow (/ h l) 1.5)
           (/ (* (pow (* D M) 2.0) -0.125) d)
           (* (sqrt (/ h l)) d))
          h)
         (*
          (/ (sqrt d) (sqrt h))
          (* (sqrt (/ d l)) (fma (* t_2 (* -0.125 (/ h l))) t_2 1.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = ((D * (0.5 / d)) * M) / l;
	double t_2 = (M / d) * D;
	double tmp;
	if (d <= -1.3e+118) {
		tmp = fma((((-D * 0.25) * (M / d)) * t_1), h, 1.0) * (pow((d / l), (1.0 / 2.0)) * (t_0 / sqrt(-h)));
	} else if (d <= -6e-175) {
		tmp = (1.0 - (((((D * 0.5) * 0.5) * (M / d)) / pow(h, -1.0)) * t_1)) * ((sqrt(pow(-l, -1.0)) * t_0) * pow((d / h), (1.0 / 2.0)));
	} else if (d <= 2.4e-182) {
		tmp = fma(pow((h / l), 1.5), ((pow((D * M), 2.0) * -0.125) / d), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * fma((t_2 * (-0.125 * (h / l))), t_2, 1.0));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(Float64(D * Float64(0.5 / d)) * M) / l)
	t_2 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (d <= -1.3e+118)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D) * 0.25) * Float64(M / d)) * t_1), h, 1.0) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(t_0 / sqrt(Float64(-h)))));
	elseif (d <= -6e-175)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * 0.5) * 0.5) * Float64(M / d)) / (h ^ -1.0)) * t_1)) * Float64(Float64(sqrt((Float64(-l) ^ -1.0)) * t_0) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	elseif (d <= 2.4e-182)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(D * M) ^ 2.0) * -0.125) / d), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(t_2 * Float64(-0.125 * Float64(h / l))), t_2, 1.0)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[d, -1.3e+118], N[(N[(N[(N[(N[((-D) * 0.25), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-175], N[(N[(1.0 - N[(N[(N[(N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[(-l), -1.0], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-182], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.30000000000000008e118

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 77.0

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

   4. Applied rewrites 77.0%

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

   6. Applied rewrites 85.8%

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

   if -1.30000000000000008e118 < d < -6e-175

   1. Initial program 78.2%

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

   4. Applied rewrites 83.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. div-inv N/A

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

      9. sqrt-prod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-neg.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if -6e-175 < d < 2.3999999999999998e-182

   1. Initial program 41.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.1%

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

   7. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 2.3999999999999998e-182 < d

   1. Initial program 77.0%

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 77.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 87.6

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

   10. Applied rewrites 87.6%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-175}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\right) \cdot \left(\left(\sqrt{{\left(-\ell\right)}^{-1}} \cdot \sqrt{-d}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(-0.125 \cdot h\right) \cdot \left(D \cdot M\right)}{\ell \cdot d}, t\_0, 1\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-247}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{d} \cdot \frac{\left(D \cdot M\right) \cdot -0.125}{\ell}, t\_0, 1\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (/ d h)))
        (t_4 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_1 -4e-172)
     (* (* (fma (/ (* (* -0.125 h) (* D M)) (* l d)) t_0 1.0) t_2) t_3)
     (if (<= t_1 1e-247)
       t_4
       (if (<= t_1 5e+157)
         (* (* (fma (* (/ h d) (/ (* (* D M) -0.125) l)) t_0 1.0) t_2) t_3)
         t_4)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double t_2 = sqrt((d / l));
	double t_3 = sqrt((d / h));
	double t_4 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_1 <= -4e-172) {
		tmp = (fma((((-0.125 * h) * (D * M)) / (l * d)), t_0, 1.0) * t_2) * t_3;
	} else if (t_1 <= 1e-247) {
		tmp = t_4;
	} else if (t_1 <= 5e+157) {
		tmp = (fma(((h / d) * (((D * M) * -0.125) / l)), t_0, 1.0) * t_2) * t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(d / h))
	t_4 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= -4e-172)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(-0.125 * h) * Float64(D * M)) / Float64(l * d)), t_0, 1.0) * t_2) * t_3);
	elseif (t_1 <= 1e-247)
		tmp = t_4;
	elseif (t_1 <= 5e+157)
		tmp = Float64(Float64(fma(Float64(Float64(h / d) * Float64(Float64(Float64(D * M) * -0.125) / l)), t_0, 1.0) * t_2) * t_3);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e-172], N[(N[(N[(N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 1e-247], t$95$4, If[LessEqual[t$95$1, 5e+157], N[(N[(N[(N[(N[(h / d), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$4]]]]]]]]

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)))) < -4.0000000000000002e-172

   1. Initial program 86.9%

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 88.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 88.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if -4.0000000000000002e-172 < (*.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-247 or 4.99999999999999976e157 < (*.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 22.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 31.1

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

   5. Applied rewrites 31.1%

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

   7. Applied rewrites 55.7%

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

   if 1e-247 < (*.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)))) < 4.99999999999999976e157

   1. Initial program 98.3%

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

   4. Applied rewrites 86.0%

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

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 98.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in h around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 98.2

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

   11. Applied rewrites 98.2%

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.125 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(-0.125 \cdot h\right) \cdot \left(D \cdot M\right)}{\ell \cdot d}, \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{d} \cdot \frac{\left(D \cdot M\right) \cdot -0.125}{\ell}, \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(-0.125 \cdot h\right) \cdot \left(D \cdot M\right)}{\ell \cdot d}, \frac{M}{d} \cdot D, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{-247}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 -4e-172)
     (*
      (* (fma (/ (* (* -0.125 h) (* D M)) (* l d)) (* (/ M d) D) 1.0) t_1)
      t_2)
     (if (<= t_0 1e-247) t_3 (if (<= t_0 5e+157) (* t_2 t_1) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= -4e-172) {
		tmp = (fma((((-0.125 * h) * (D * M)) / (l * d)), ((M / d) * D), 1.0) * t_1) * t_2;
	} else if (t_0 <= 1e-247) {
		tmp = t_3;
	} else if (t_0 <= 5e+157) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -4e-172)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(-0.125 * h) * Float64(D * M)) / Float64(l * d)), Float64(Float64(M / d) * D), 1.0) * t_1) * t_2);
	elseif (t_0 <= 1e-247)
		tmp = t_3;
	elseif (t_0 <= 5e+157)
		tmp = Float64(t_2 * t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-172], N[(N[(N[(N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e-247], t$95$3, If[LessEqual[t$95$0, 5e+157], N[(t$95$2 * t$95$1), $MachinePrecision], t$95$3]]]]]]]

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)))) < -4.0000000000000002e-172

   1. Initial program 86.9%

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 88.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 88.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if -4.0000000000000002e-172 < (*.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-247 or 4.99999999999999976e157 < (*.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 22.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 31.1

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

   5. Applied rewrites 31.1%

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

   7. Applied rewrites 55.7%

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

   if 1e-247 < (*.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)))) < 4.99999999999999976e157

   1. Initial program 98.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(-0.125 \cdot h\right) \cdot \left(D \cdot M\right)}{\ell \cdot d}, \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \frac{D \cdot D}{d}\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))))
        (t_1 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 -4e-172)
     (* (* (/ (sqrt (/ h l)) (fabs l)) (/ (* D D) d)) (* (* M M) -0.125))
     (if (<= t_0 1e-247)
       t_1
       (if (<= t_0 5e+157) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double t_1 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= -4e-172) {
		tmp = ((sqrt((h / l)) / fabs(l)) * ((D * D) / d)) * ((M * M) * -0.125);
	} else if (t_0 <= 1e-247) {
		tmp = t_1;
	} else if (t_0 <= 5e+157) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    t_1 = abs((d / sqrt((l * h))))
    if (t_0 <= (-4d-172)) then
        tmp = ((sqrt((h / l)) / abs(l)) * ((d_1 * d_1) / d)) * ((m * m) * (-0.125d0))
    else if (t_0 <= 1d-247) then
        tmp = t_1
    else if (t_0 <= 5d+157) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double t_1 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_0 <= -4e-172) {
		tmp = ((Math.sqrt((h / l)) / Math.abs(l)) * ((D * D) / d)) * ((M * M) * -0.125);
	} else if (t_0 <= 1e-247) {
		tmp = t_1;
	} else if (t_0 <= 5e+157) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	t_1 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_0 <= -4e-172:
		tmp = ((math.sqrt((h / l)) / math.fabs(l)) * ((D * D) / d)) * ((M * M) * -0.125)
	elif t_0 <= 1e-247:
		tmp = t_1
	elif t_0 <= 5e+157:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	t_1 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -4e-172)
		tmp = Float64(Float64(Float64(sqrt(Float64(h / l)) / abs(l)) * Float64(Float64(D * D) / d)) * Float64(Float64(M * M) * -0.125));
	elseif (t_0 <= 1e-247)
		tmp = t_1;
	elseif (t_0 <= 5e+157)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	t_1 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_0 <= -4e-172)
		tmp = ((sqrt((h / l)) / abs(l)) * ((D * D) / d)) * ((M * M) * -0.125);
	elseif (t_0 <= 1e-247)
		tmp = t_1;
	elseif (t_0 <= 5e+157)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-172], N[(N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-247], t$95$1, If[LessEqual[t$95$0, 5e+157], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))) < -4.0000000000000002e-172

   1. Initial program 86.9%

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

   3. Taylor expanded in h around inf

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 39.9%

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

   7. Applied rewrites 46.0%

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

   if -4.0000000000000002e-172 < (*.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-247 or 4.99999999999999976e157 < (*.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 22.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 31.1

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

   5. Applied rewrites 31.1%

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

   7. Applied rewrites 55.7%

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

   if 1e-247 < (*.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)))) < 4.99999999999999976e157

   1. Initial program 98.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \frac{D \cdot D}{d}\right) \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{elif}\;t\_1 \leq 10^{-247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* l h)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 -4e-172)
     (/ (* (sqrt (/ h l)) (- d)) h)
     (if (<= t_1 1e-247)
       t_0
       (if (<= t_1 5e+157) (* (sqrt (/ d h)) (sqrt (/ d l))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h))));
	double t_1 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -4e-172) {
		tmp = (sqrt((h / l)) * -d) / h;
	} else if (t_1 <= 1e-247) {
		tmp = t_0;
	} else if (t_1 <= 5e+157) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((l * h))))
    t_1 = (1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-4d-172)) then
        tmp = (sqrt((h / l)) * -d) / h
    else if (t_1 <= 1d-247) then
        tmp = t_0
    else if (t_1 <= 5d+157) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h))));
	double t_1 = (1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -4e-172) {
		tmp = (Math.sqrt((h / l)) * -d) / h;
	} else if (t_1 <= 1e-247) {
		tmp = t_0;
	} else if (t_1 <= 5e+157) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h))))
	t_1 = (1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -4e-172:
		tmp = (math.sqrt((h / l)) * -d) / h
	elif t_1 <= 1e-247:
		tmp = t_0
	elif t_1 <= 5e+157:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(l * h))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -4e-172)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d)) / h);
	elseif (t_1 <= 1e-247)
		tmp = t_0;
	elseif (t_1 <= 5e+157)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h))));
	t_1 = (1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -4e-172)
		tmp = (sqrt((h / l)) * -d) / h;
	elseif (t_1 <= 1e-247)
		tmp = t_0;
	elseif (t_1 <= 5e+157)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-172], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 1e-247], t$95$0, If[LessEqual[t$95$1, 5e+157], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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)))) < -4.0000000000000002e-172

   1. Initial program 86.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 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. 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 29.6%

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

   if -4.0000000000000002e-172 < (*.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-247 or 4.99999999999999976e157 < (*.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 22.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 31.1

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

   5. Applied rewrites 31.1%

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

   7. Applied rewrites 55.7%

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

   if 1e-247 < (*.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)))) < 4.99999999999999976e157

   1. Initial program 98.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_2 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{t\_0 \cdot \left(-d\right)}{h}\\ \mathbf{elif}\;t\_2 \leq 10^{-247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+196}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1 (fabs (/ d (sqrt (* l h)))))
        (t_2
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_2 -4e-172)
     (/ (* t_0 (- d)) h)
     (if (<= t_2 1e-247) t_1 (if (<= t_2 1e+196) (/ (* t_0 d) h) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = fabs((d / sqrt((l * h))));
	double t_2 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_2 <= -4e-172) {
		tmp = (t_0 * -d) / h;
	} else if (t_2 <= 1e-247) {
		tmp = t_1;
	} else if (t_2 <= 1e+196) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((h / l))
    t_1 = abs((d / sqrt((l * h))))
    t_2 = (1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    if (t_2 <= (-4d-172)) then
        tmp = (t_0 * -d) / h
    else if (t_2 <= 1d-247) then
        tmp = t_1
    else if (t_2 <= 1d+196) then
        tmp = (t_0 * d) / h
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = Math.abs((d / Math.sqrt((l * h))));
	double t_2 = (1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_2 <= -4e-172) {
		tmp = (t_0 * -d) / h;
	} else if (t_2 <= 1e-247) {
		tmp = t_1;
	} else if (t_2 <= 1e+196) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = math.fabs((d / math.sqrt((l * h))))
	t_2 = (1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_2 <= -4e-172:
		tmp = (t_0 * -d) / h
	elif t_2 <= 1e-247:
		tmp = t_1
	elif t_2 <= 1e+196:
		tmp = (t_0 * d) / h
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = abs(Float64(d / sqrt(Float64(l * h))))
	t_2 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_2 <= -4e-172)
		tmp = Float64(Float64(t_0 * Float64(-d)) / h);
	elseif (t_2 <= 1e-247)
		tmp = t_1;
	elseif (t_2 <= 1e+196)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = abs((d / sqrt((l * h))));
	t_2 = (1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_2 <= -4e-172)
		tmp = (t_0 * -d) / h;
	elseif (t_2 <= 1e-247)
		tmp = t_1;
	elseif (t_2 <= 1e+196)
		tmp = (t_0 * d) / h;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-172], N[(N[(t$95$0 * (-d)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 1e-247], t$95$1, If[LessEqual[t$95$2, 1e+196], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$1]]]]]]

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)))) < -4.0000000000000002e-172

   1. Initial program 86.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 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. 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 29.6%

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

   if -4.0000000000000002e-172 < (*.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-247 or 9.9999999999999995e195 < (*.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 20.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 29.9

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

   5. Applied rewrites 29.9%

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

   7. Applied rewrites 54.2%

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

   if 1e-247 < (*.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)))) < 9.9999999999999995e195

   1. Initial program 98.3%

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

   3. Taylor expanded in 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 69.4%

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

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 92.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+196}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+196}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* l h)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 -5e+51)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= t_1 1e-247)
       t_0
       (if (<= t_1 1e+196) (/ (* (sqrt (/ h l)) d) h) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h))));
	double t_1 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 1e-247) {
		tmp = t_0;
	} else if (t_1 <= 1e+196) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((l * h))))
    t_1 = (1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-5d+51)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else if (t_1 <= 1d-247) then
        tmp = t_0
    else if (t_1 <= 1d+196) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h))));
	double t_1 = (1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 1e-247) {
		tmp = t_0;
	} else if (t_1 <= 1e+196) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h))))
	t_1 = (1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	elif t_1 <= 1e-247:
		tmp = t_0
	elif t_1 <= 1e+196:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(l * h))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (t_1 <= 1e-247)
		tmp = t_0;
	elseif (t_1 <= 1e+196)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h))));
	t_1 = (1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = sqrt((1.0 / (l * h))) * -d;
	elseif (t_1 <= 1e-247)
		tmp = t_0;
	elseif (t_1 <= 1e+196)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+51], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 1e-247], t$95$0, If[LessEqual[t$95$1, 1e+196], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], t$95$0]]]]]

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)))) < -5e51

   1. Initial program 86.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. 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 14.1

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

   5. Applied rewrites 14.1%

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

   if -5e51 < (*.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-247 or 9.9999999999999995e195 < (*.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 22.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 29.1

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

   5. Applied rewrites 29.1%

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

   7. Applied rewrites 52.6%

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

   if 1e-247 < (*.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)))) < 9.9999999999999995e195

   1. Initial program 98.3%

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

   3. Taylor expanded in 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 69.4%

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

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 92.5%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+196}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* l h)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 -5e+51)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= t_1 2e-200)
       t_0
       (if (<= t_1 4e+97) (sqrt (* (/ (/ d l) h) d)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h))));
	double t_1 = (1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 2e-200) {
		tmp = t_0;
	} else if (t_1 <= 4e+97) {
		tmp = sqrt((((d / l) / h) * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((l * h))))
    t_1 = (1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-5d+51)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else if (t_1 <= 2d-200) then
        tmp = t_0
    else if (t_1 <= 4d+97) then
        tmp = sqrt((((d / l) / h) * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h))));
	double t_1 = (1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 2e-200) {
		tmp = t_0;
	} else if (t_1 <= 4e+97) {
		tmp = Math.sqrt((((d / l) / h) * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h))))
	t_1 = (1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	elif t_1 <= 2e-200:
		tmp = t_0
	elif t_1 <= 4e+97:
		tmp = math.sqrt((((d / l) / h) * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(l * h))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (t_1 <= 2e-200)
		tmp = t_0;
	elseif (t_1 <= 4e+97)
		tmp = sqrt(Float64(Float64(Float64(d / l) / h) * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h))));
	t_1 = (1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = sqrt((1.0 / (l * h))) * -d;
	elseif (t_1 <= 2e-200)
		tmp = t_0;
	elseif (t_1 <= 4e+97)
		tmp = sqrt((((d / l) / h) * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+51], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 2e-200], t$95$0, If[LessEqual[t$95$1, 4e+97], N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

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)))) < -5e51

   1. Initial program 86.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. 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 14.1

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

   5. Applied rewrites 14.1%

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

   if -5e51 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e-200 or 4.0000000000000003e97 < (*.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 35.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   7. Applied rewrites 56.9%

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

   if 2e-200 < (*.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)))) < 4.0000000000000003e97

   1. Initial program 99.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   7. Applied rewrites 96.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-200}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D \cdot M}{d} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      5e+157)
   (*
    (*
     (fma (* (/ (* D M) d) (* -0.125 (/ h l))) (* (/ M d) D) 1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 5e+157) {
		tmp = (fma((((D * M) / d) * (-0.125 * (h / l))), ((M / d) * D), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 5e+157)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(D * M) / d) * Float64(-0.125 * Float64(h / l))), Float64(Float64(M / d) * D), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 5e+157], N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e157

   1. Initial program 86.8%

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 87.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.7

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

   10. Applied rewrites 87.7%

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

   if 4.99999999999999976e157 < (*.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 18.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 24.0

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

   5. Applied rewrites 24.0%

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

   7. Applied rewrites 48.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D \cdot M}{d} \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D)))
   (if (<=
        (*
         (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
        5e+157)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0)))
     (fabs (/ d (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 5e+157) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma((t_0 * (-0.125 * (h / l))), t_0, 1.0));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 5e+157)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 5e+157], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e157

   1. Initial program 86.8%

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 87.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if 4.99999999999999976e157 < (*.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 18.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 24.0

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

   5. Applied rewrites 24.0%

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

   7. Applied rewrites 48.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{D}{d} \cdot M, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      5e+157)
   (*
    (*
     (fma (* (* (/ M d) D) (* -0.125 (/ h l))) (* (/ D d) M) 1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 5e+157) {
		tmp = (fma((((M / d) * D) * (-0.125 * (h / l))), ((D / d) * M), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 5e+157)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M / d) * D) * Float64(-0.125 * Float64(h / l))), Float64(Float64(D / d) * M), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 5e+157], N[(N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e157

   1. Initial program 86.8%

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 87.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if 4.99999999999999976e157 < (*.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 18.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 24.0

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

   5. Applied rewrites 24.0%

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

   7. Applied rewrites 48.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{D}{d} \cdot M, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -5e+51)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (fabs (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -5e+51) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= (-5d+51)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= -5e+51) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= -5e+51:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -5e+51)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= -5e+51)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -5e+51], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e51

   1. Initial program 86.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. 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 14.1

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

   5. Applied rewrites 14.1%

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

   if -5e51 < (*.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 54.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   7. Applied rewrites 60.8%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -2e-160)
   (* (sqrt (/ 1.0 (* l h))) d)
   (fabs (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -2e-160) {
		tmp = sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= (-2d-160)) then
        tmp = sqrt((1.0d0 / (l * h))) * d
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= -2e-160) {
		tmp = Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= -2e-160:
		tmp = math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -2e-160)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= -2e-160)
		tmp = sqrt((1.0 / (l * h))) * d;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -2e-160], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-160

   1. Initial program 86.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 11.9

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

   5. Applied rewrites 11.9%

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

   if -2e-160 < (*.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 53.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 61.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<=
        (*
         (- 1.0 (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
        -2e-160)
     t_0
     (fabs t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (((1.0 - ((pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -2e-160) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (((1.0d0 - (((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= (-2d-160)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (((1.0 - ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= -2e-160) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if ((1.0 - ((math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= -2e-160:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -2e-160)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (((1.0 - (((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= -2e-160)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -2e-160], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-160

   1. Initial program 86.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 11.9

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 11.0%

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

   if -2e-160 < (*.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 53.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 61.6%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\\ t_2 := \frac{M}{d} \cdot D\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot t\_1, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{t\_0}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot t\_1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t\_2 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_2, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (/ (* (* D (/ 0.5 d)) M) l))
        (t_2 (* (/ M d) D)))
   (if (<= d -1.35e+118)
     (*
      (fma (* (* (* (- D) 0.25) (/ M d)) t_1) h 1.0)
      (* (pow (/ d l) (/ 1.0 2.0)) (/ t_0 (sqrt (- h)))))
     (if (<= d -6e-175)
       (*
        (* (/ t_0 (sqrt (- l))) (pow (/ d h) (/ 1.0 2.0)))
        (- 1.0 (* (/ (* (* (* D 0.5) 0.5) (/ M d)) (pow h -1.0)) t_1)))
       (if (<= d 2.4e-182)
         (/
          (fma
           (pow (/ h l) 1.5)
           (/ (* (pow (* D M) 2.0) -0.125) d)
           (* (sqrt (/ h l)) d))
          h)
         (*
          (/ (sqrt d) (sqrt h))
          (* (sqrt (/ d l)) (fma (* t_2 (* -0.125 (/ h l))) t_2 1.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = ((D * (0.5 / d)) * M) / l;
	double t_2 = (M / d) * D;
	double tmp;
	if (d <= -1.35e+118) {
		tmp = fma((((-D * 0.25) * (M / d)) * t_1), h, 1.0) * (pow((d / l), (1.0 / 2.0)) * (t_0 / sqrt(-h)));
	} else if (d <= -6e-175) {
		tmp = ((t_0 / sqrt(-l)) * pow((d / h), (1.0 / 2.0))) * (1.0 - (((((D * 0.5) * 0.5) * (M / d)) / pow(h, -1.0)) * t_1));
	} else if (d <= 2.4e-182) {
		tmp = fma(pow((h / l), 1.5), ((pow((D * M), 2.0) * -0.125) / d), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * fma((t_2 * (-0.125 * (h / l))), t_2, 1.0));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(Float64(D * Float64(0.5 / d)) * M) / l)
	t_2 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (d <= -1.35e+118)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D) * 0.25) * Float64(M / d)) * t_1), h, 1.0) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(t_0 / sqrt(Float64(-h)))));
	elseif (d <= -6e-175)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-l))) * (Float64(d / h) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * 0.5) * 0.5) * Float64(M / d)) / (h ^ -1.0)) * t_1)));
	elseif (d <= 2.4e-182)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(D * M) ^ 2.0) * -0.125) / d), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(t_2 * Float64(-0.125 * Float64(h / l))), t_2, 1.0)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[d, -1.35e+118], N[(N[(N[(N[(N[((-D) * 0.25), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-175], N[(N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-182], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.35e118

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 77.0

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

   4. Applied rewrites 77.0%

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

   6. Applied rewrites 85.8%

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

   if -1.35e118 < d < -6e-175

   1. Initial program 78.2%

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

   4. Applied rewrites 83.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if -6e-175 < d < 2.3999999999999998e-182

   1. Initial program 41.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.1%

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

   7. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 2.3999999999999998e-182 < d

   1. Initial program 77.0%

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 77.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 87.6

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

   10. Applied rewrites 87.6%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D)))
   (if (<= d -7.8e-174)
     (*
      (fma (* (* (* (- D) 0.25) (/ M d)) (/ (* (* D (/ 0.5 d)) M) l)) h 1.0)
      (* (pow (/ d l) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d 2.4e-182)
       (/
        (fma
         (pow (/ h l) 1.5)
         (/ (* (pow (* D M) 2.0) -0.125) d)
         (* (sqrt (/ h l)) d))
        h)
       (*
        (/ (sqrt d) (sqrt h))
        (* (sqrt (/ d l)) (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double tmp;
	if (d <= -7.8e-174) {
		tmp = fma((((-D * 0.25) * (M / d)) * (((D * (0.5 / d)) * M) / l)), h, 1.0) * (pow((d / l), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= 2.4e-182) {
		tmp = fma(pow((h / l), 1.5), ((pow((D * M), 2.0) * -0.125) / d), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * fma((t_0 * (-0.125 * (h / l))), t_0, 1.0));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (d <= -7.8e-174)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D) * 0.25) * Float64(M / d)) * Float64(Float64(Float64(D * Float64(0.5 / d)) * M) / l)), h, 1.0) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= 2.4e-182)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(D * M) ^ 2.0) * -0.125) / d), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[d, -7.8e-174], N[(N[(N[(N[(N[((-D) * 0.25), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-182], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.7999999999999997e-174

   1. Initial program 70.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.8

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

   4. Applied rewrites 78.8%

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

   6. Applied rewrites 84.3%

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

   if -7.7999999999999997e-174 < d < 2.3999999999999998e-182

   1. Initial program 41.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.1%

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

   7. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 2.3999999999999998e-182 < d

   1. Initial program 77.0%

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 77.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 87.6

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

   10. Applied rewrites 87.6%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D)) (t_1 (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0)))
   (if (<= l -1e-311)
     (*
      (fma (* (* (* (- D) 0.25) (/ M d)) (/ (* (* D (/ 0.5 d)) M) l)) h 1.0)
      (* (pow (/ d l) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= l 2.6e+171)
       (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_1))
       (* (sqrt (/ d h)) (* (/ (sqrt d) (sqrt l)) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = fma((t_0 * (-0.125 * (h / l))), t_0, 1.0);
	double tmp;
	if (l <= -1e-311) {
		tmp = fma((((-D * 0.25) * (M / d)) * (((D * (0.5 / d)) * M) / l)), h, 1.0) * (pow((d / l), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-h)));
	} else if (l <= 2.6e+171) {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_1);
	} else {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * t_1);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D) * 0.25) * Float64(M / d)) * Float64(Float64(Float64(D * Float64(0.5 / d)) * M) / l)), h, 1.0) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (l <= 2.6e+171)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * t_1));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(N[(N[(N[(N[((-D) * 0.25), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+171], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 63.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 71.6

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

   4. Applied rewrites 71.6%

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

   6. Applied rewrites 76.0%

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

   if -9.99999999999948e-312 < l < 2.6e171

   1. Initial program 74.0%

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

   4. Applied rewrites 50.0%

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

   6. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 74.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if 2.6e171 < l

   1. Initial program 51.1%

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

   4. Applied rewrites 37.4%

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

   6. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 53.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 53.7

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

   8. Applied rewrites 53.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 75.8

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

   10. Applied rewrites 75.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot 0.25\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}, h, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;\left(\left({\left(-\ell\right)}^{-0.5} \cdot t\_2\right) \cdot t\_1\right) \cdot t\_3\\ \mathbf{elif}\;\ell \leq -5.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right) \cdot t\_4\right) \cdot t\_2}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_4 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0))
        (t_2 (sqrt (- d)))
        (t_3 (sqrt (/ d h)))
        (t_4 (sqrt (/ d l))))
   (if (<= l -2.85e+163)
     (* (* (* (pow (- l) -0.5) t_2) t_1) t_3)
     (if (<= l -5.7e-304)
       (/
        (*
         (* (fma (* -0.5 (/ h l)) (pow (* (/ d D) (/ 2.0 M)) -2.0) 1.0) t_4)
         t_2)
        (sqrt (- h)))
       (if (<= l 2.6e+171)
         (* (/ (sqrt d) (sqrt h)) (* t_4 t_1))
         (* t_3 (* (/ (sqrt d) (sqrt l)) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = fma((t_0 * (-0.125 * (h / l))), t_0, 1.0);
	double t_2 = sqrt(-d);
	double t_3 = sqrt((d / h));
	double t_4 = sqrt((d / l));
	double tmp;
	if (l <= -2.85e+163) {
		tmp = ((pow(-l, -0.5) * t_2) * t_1) * t_3;
	} else if (l <= -5.7e-304) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / D) * (2.0 / M)), -2.0), 1.0) * t_4) * t_2) / sqrt(-h);
	} else if (l <= 2.6e+171) {
		tmp = (sqrt(d) / sqrt(h)) * (t_4 * t_1);
	} else {
		tmp = t_3 * ((sqrt(d) / sqrt(l)) * t_1);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)
	t_2 = sqrt(Float64(-d))
	t_3 = sqrt(Float64(d / h))
	t_4 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.85e+163)
		tmp = Float64(Float64(Float64((Float64(-l) ^ -0.5) * t_2) * t_1) * t_3);
	elseif (l <= -5.7e-304)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / D) * Float64(2.0 / M)) ^ -2.0), 1.0) * t_4) * t_2) / sqrt(Float64(-h)));
	elseif (l <= 2.6e+171)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_4 * t_1));
	else
		tmp = Float64(t_3 * Float64(Float64(sqrt(d) / sqrt(l)) * t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.85e+163], N[(N[(N[(N[Power[(-l), -0.5], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[l, -5.7e-304], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / D), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+171], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.8499999999999999e163

   1. Initial program 33.9%

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

   4. Applied rewrites 30.7%

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

   6. Applied rewrites 33.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 39.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 39.7

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

   8. Applied rewrites 39.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. inv-pow N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-pow2 N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 51.0

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \left(\sqrt{-d} \cdot \color{blue}{{\left(-\ell\right)}^{-0.5}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   10. Applied rewrites 51.0%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\left(\sqrt{-d} \cdot {\left(-\ell\right)}^{-0.5}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -2.8499999999999999e163 < l < -5.6999999999999998e-304

   1. Initial program 76.5%

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

   4. Applied rewrites 86.7%

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

   if -5.6999999999999998e-304 < l < 2.6e171

   1. Initial program 73.3%

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

   4. Applied rewrites 49.6%

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

   6. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 73.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 85.6

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

   10. Applied rewrites 85.6%

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

   if 2.6e171 < l

   1. Initial program 51.1%

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

   4. Applied rewrites 37.4%

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

   6. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 53.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 53.7

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

   8. Applied rewrites 53.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 75.8

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

   10. Applied rewrites 75.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;\left(\left({\left(-\ell\right)}^{-0.5} \cdot \sqrt{-d}\right) \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\left({\left(-\ell\right)}^{-0.5} \cdot \sqrt{-d}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0))
        (t_2 (sqrt (/ d h))))
   (if (<= l -1e-311)
     (* (* (* (pow (- l) -0.5) (sqrt (- d))) t_1) t_2)
     (if (<= l 2.6e+171)
       (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_1))
       (* t_2 (* (/ (sqrt d) (sqrt l)) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = fma((t_0 * (-0.125 * (h / l))), t_0, 1.0);
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= -1e-311) {
		tmp = ((pow(-l, -0.5) * sqrt(-d)) * t_1) * t_2;
	} else if (l <= 2.6e+171) {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_1);
	} else {
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * t_1);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(Float64((Float64(-l) ^ -0.5) * sqrt(Float64(-d))) * t_1) * t_2);
	elseif (l <= 2.6e+171)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * t_1));
	else
		tmp = Float64(t_2 * Float64(Float64(sqrt(d) / sqrt(l)) * t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(N[(N[(N[Power[(-l), -0.5], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 2.6e+171], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 63.8%

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

   4. Applied rewrites 40.6%

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

   6. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 65.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.5

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

   8. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. inv-pow N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-pow2 N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 70.6

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \left(\sqrt{-d} \cdot \color{blue}{{\left(-\ell\right)}^{-0.5}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   10. Applied rewrites 70.6%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\left(\sqrt{-d} \cdot {\left(-\ell\right)}^{-0.5}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -9.99999999999948e-312 < l < 2.6e171

   1. Initial program 74.0%

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

   4. Applied rewrites 50.0%

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

   6. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 74.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if 2.6e171 < l

   1. Initial program 51.1%

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

   4. Applied rewrites 37.4%

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

   6. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 53.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 53.7

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

   8. Applied rewrites 53.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 75.8

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

   10. Applied rewrites 75.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\left({\left(-\ell\right)}^{-0.5} \cdot \sqrt{-d}\right) \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 75.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (fma (* t_0 (* -0.125 (/ h l))) t_0 1.0))
        (t_2 (sqrt (/ d h))))
   (if (<= l -1e-311)
     (* (* (/ (sqrt (- d)) (sqrt (- l))) t_1) t_2)
     (if (<= l 2.6e+171)
       (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_1))
       (* t_2 (* (/ (sqrt d) (sqrt l)) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = fma((t_0 * (-0.125 * (h / l))), t_0, 1.0);
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= -1e-311) {
		tmp = ((sqrt(-d) / sqrt(-l)) * t_1) * t_2;
	} else if (l <= 2.6e+171) {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_1);
	} else {
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * t_1);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = fma(Float64(t_0 * Float64(-0.125 * Float64(h / l))), t_0, 1.0)
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * t_1) * t_2);
	elseif (l <= 2.6e+171)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * t_1));
	else
		tmp = Float64(t_2 * Float64(Float64(sqrt(d) / sqrt(l)) * t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 2.6e+171], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 63.8%

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

   4. Applied rewrites 40.6%

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

   6. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 65.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. lower-*.f64 65.5

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{D \cdot \frac{M}{d}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. lower-*.f64 65.5

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   8. Applied rewrites 65.5%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. frac-2neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{-d}{\color{blue}{-\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\color{blue}{{\left(-d\right)}^{\frac{1}{2}}}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{{\left(-d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{{\left(-d\right)}^{\left(\frac{1}{2}\right)}}{\color{blue}{\sqrt{-\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{{\left(-d\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{{\left(-d\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. pow1/2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. lower-sqrt.f64 70.4

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   10. Applied rewrites 70.4%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -9.99999999999948e-312 < l < 2.6e171

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(-0.5 \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}, \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\right)} \]

   6. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) + 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{1}{4}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{-1}{8}}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. lower-*.f64 74.0

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot -0.125\right)} \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. lower-*.f64 74.0

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{D \cdot \frac{M}{d}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. lower-*.f64 74.0

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   8. Applied rewrites 74.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\color{blue}{\frac{d}{h}}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

      4. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{{h}^{\frac{1}{2}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{{h}^{\left(\frac{1}{2}\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{{h}^{\color{blue}{\frac{1}{2}}}} \]

      9. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \]

      10. lower-sqrt.f64 86.4

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \]

   10. Applied rewrites 86.4%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

   if 2.6e171 < l

   1. Initial program 51.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(-0.5 \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}, \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\right)} \]

   6. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) + 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{1}{4}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{-1}{8}}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. lower-*.f64 53.7

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot -0.125\right)} \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. lower-*.f64 53.7

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{D \cdot \frac{M}{d}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. lower-*.f64 53.7

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \color{blue}{\frac{M}{d} \cdot D}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   8. Applied rewrites 53.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\sqrt{d}}{\color{blue}{{\ell}^{\frac{1}{2}}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\sqrt{d}}{{\ell}^{\color{blue}{\left(\frac{1}{2}\right)}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{{\ell}^{\left(\frac{1}{2}\right)}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\color{blue}{\sqrt{d}}}{{\ell}^{\left(\frac{1}{2}\right)}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\sqrt{d}}{{\ell}^{\color{blue}{\frac{1}{2}}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{8}\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. lower-sqrt.f64 75.8

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

   10. Applied rewrites 75.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(\frac{M}{d} \cdot D\right), \frac{M}{d} \cdot D, 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.125 \cdot \frac{h}{\ell}\right), \frac{M}{d} \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 46.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 4.5e-245)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4.5e-245) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 4.5d-245) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4.5e-245) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 4.5e-245:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 4.5e-245)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 4.5e-245)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 4.5e-245], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.49999999999999969e-245

   1. Initial program 64.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 37.7

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.49999999999999969e-245 < l

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 42.7

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 27.3

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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)))))