Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.6% → 81.1%

Time: 15.3s

Alternatives: 15

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.6% 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: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := \frac{D}{d} \cdot M\_m\\ t_2 := \mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot t\_1\right)}{\ell} \cdot M\_m, 0.5 \cdot \frac{D}{d}, 1\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(t\_1 \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \left({\left(-h\right)}^{-0.5} \cdot \sqrt{-d}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.9 \cdot 10^{-96}:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(D \cdot M\_m\right)}^{2}}{d}, -0.125 \cdot \left({\ell}^{-1.5} \cdot \sqrt{h}\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{h}} \cdot t\_2\right) \cdot \sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h))))
        (t_1 (* (/ D d) M_m))
        (t_2 (fma (* (/ (* h (* -0.25 t_1)) l) M_m) (* 0.5 (/ D d)) 1.0)))
   (if (<= l -5e-310)
     (*
      (*
       (fma (* (* M_m (/ D d)) 0.5) (/ (* (* t_1 -0.25) h) l) 1.0)
       (* (pow (- h) -0.5) (sqrt (- d))))
      (sqrt (/ d l)))
     (if (<= l 6.9e-96)
       (* t_2 t_0)
       (if (<= l 1.75e+130)
         (fma
          (/ (pow (* D M_m) 2.0) d)
          (* -0.125 (* (pow l -1.5) (sqrt h)))
          t_0)
         (/ (* (* (sqrt (/ d h)) t_2) (sqrt d)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double t_1 = (D / d) * M_m;
	double t_2 = fma((((h * (-0.25 * t_1)) / l) * M_m), (0.5 * (D / d)), 1.0);
	double tmp;
	if (l <= -5e-310) {
		tmp = (fma(((M_m * (D / d)) * 0.5), (((t_1 * -0.25) * h) / l), 1.0) * (pow(-h, -0.5) * sqrt(-d))) * sqrt((d / l));
	} else if (l <= 6.9e-96) {
		tmp = t_2 * t_0;
	} else if (l <= 1.75e+130) {
		tmp = fma((pow((D * M_m), 2.0) / d), (-0.125 * (pow(l, -1.5) * sqrt(h))), t_0);
	} else {
		tmp = ((sqrt((d / h)) * t_2) * sqrt(d)) / sqrt(l);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(Float64(D / d) * M_m)
	t_2 = fma(Float64(Float64(Float64(h * Float64(-0.25 * t_1)) / l) * M_m), Float64(0.5 * Float64(D / d)), 1.0)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) * 0.5), Float64(Float64(Float64(t_1 * -0.25) * h) / l), 1.0) * Float64((Float64(-h) ^ -0.5) * sqrt(Float64(-d)))) * sqrt(Float64(d / l)));
	elseif (l <= 6.9e-96)
		tmp = Float64(t_2 * t_0);
	elseif (l <= 1.75e+130)
		tmp = fma(Float64((Float64(D * M_m) ^ 2.0) / d), Float64(-0.125 * Float64((l ^ -1.5) * sqrt(h))), t_0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(d / h)) * t_2) * sqrt(d)) / sqrt(l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(h * N[(-0.25 * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(t$95$1 * -0.25), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[(-h), -0.5], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.9e-96], N[(t$95$2 * t$95$0), $MachinePrecision], If[LessEqual[l, 1.75e+130], N[(N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * N[(-0.125 * N[(N[Power[l, -1.5], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 72.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 77.0

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

   8. Applied rewrites 77.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. pow-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 84.0

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

   10. Applied rewrites 84.0%

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

   if -4.999999999999985e-310 < l < 6.9e-96

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 75.9

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

   8. Applied rewrites 75.9%

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

   10. Applied rewrites 84.9%

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

   if 6.9e-96 < l < 1.75e130

   1. Initial program 53.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in l around inf

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

   8. Applied rewrites 66.8%

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

   10. Applied rewrites 91.3%

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

   if 1.75e130 < l

   1. Initial program 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 47.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 49.0

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

   8. Applied rewrites 49.0%

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

   10. Applied rewrites 69.3%

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

Alternative 2: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{D}{d} \cdot D\right) \cdot h\right) \cdot \left(\frac{M\_m \cdot M\_m}{\ell} \cdot -0.125\right)}{d} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -4e-71)
     (* (* (/ (* (* (* (/ D d) D) h) (* (/ (* M_m M_m) l) -0.125)) d) t_1) t_2)
     (if (<= t_0 1e+249) (* t_2 t_1) (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-71) {
		tmp = ((((((D / d) * D) * h) * (((M_m * M_m) / l) * -0.125)) / d) * t_1) * t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-4d-71)) then
        tmp = ((((((d_1 / d) * d_1) * h) * (((m_m * m_m) / l) * (-0.125d0))) / d) * t_1) * t_2
    else if (t_0 <= 1d+249) then
        tmp = t_2 * t_1
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-71) {
		tmp = ((((((D / d) * D) * h) * (((M_m * M_m) / l) * -0.125)) / d) * t_1) * t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -4e-71:
		tmp = ((((((D / d) * D) * h) * (((M_m * M_m) / l) * -0.125)) / d) * t_1) * t_2
	elif t_0 <= 1e+249:
		tmp = t_2 * t_1
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -4e-71)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(D / d) * D) * h) * Float64(Float64(Float64(M_m * M_m) / l) * -0.125)) / d) * t_1) * t_2);
	elseif (t_0 <= 1e+249)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -4e-71)
		tmp = ((((((D / d) * D) * h) * (((M_m * M_m) / l) * -0.125)) / d) * t_1) * t_2;
	elseif (t_0 <= 1e+249)
		tmp = t_2 * t_1;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-71], N[(N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.9999999999999997e-71

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 68.0%

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

   if -3.9999999999999997e-71 < (*.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)))) < 9.9999999999999992e248

   1. Initial program 86.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

   7. Applied rewrites 81.7%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{D}{d} \cdot D\right) \cdot h\right) \cdot \left(\frac{M \cdot M}{\ell} \cdot -0.125\right)}{d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell \cdot d}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -4e-71)
     (*
      (* (fma (/ h (* l d)) (* (* -0.125 (* D D)) (/ (* M_m M_m) d)) 1.0) t_1)
      t_2)
     (if (<= t_0 1e+249) (* t_2 t_1) (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-71) {
		tmp = (fma((h / (l * d)), ((-0.125 * (D * D)) * ((M_m * M_m) / d)), 1.0) * t_1) * t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -4e-71)
		tmp = Float64(Float64(fma(Float64(h / Float64(l * d)), Float64(Float64(-0.125 * Float64(D * D)) * Float64(Float64(M_m * M_m) / d)), 1.0) * t_1) * t_2);
	elseif (t_0 <= 1e+249)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-71], N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.9999999999999997e-71

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. associate-*r/ N/A

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

   9. Applied rewrites 68.6%

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

   if -3.9999999999999997e-71 < (*.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)))) < 9.9999999999999992e248

   1. Initial program 86.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

   7. Applied rewrites 81.7%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell \cdot d}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -4e-71)
     (* (* (* (/ h (* l d)) (* (* -0.125 (* D D)) (/ (* M_m M_m) d))) t_1) t_2)
     (if (<= t_0 1e+249) (* t_2 t_1) (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-71) {
		tmp = (((h / (l * d)) * ((-0.125 * (D * D)) * ((M_m * M_m) / d))) * t_1) * t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-4d-71)) then
        tmp = (((h / (l * d)) * (((-0.125d0) * (d_1 * d_1)) * ((m_m * m_m) / d))) * t_1) * t_2
    else if (t_0 <= 1d+249) then
        tmp = t_2 * t_1
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-71) {
		tmp = (((h / (l * d)) * ((-0.125 * (D * D)) * ((M_m * M_m) / d))) * t_1) * t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -4e-71:
		tmp = (((h / (l * d)) * ((-0.125 * (D * D)) * ((M_m * M_m) / d))) * t_1) * t_2
	elif t_0 <= 1e+249:
		tmp = t_2 * t_1
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -4e-71)
		tmp = Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(Float64(-0.125 * Float64(D * D)) * Float64(Float64(M_m * M_m) / d))) * t_1) * t_2);
	elseif (t_0 <= 1e+249)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -4e-71)
		tmp = (((h / (l * d)) * ((-0.125 * (D * D)) * ((M_m * M_m) / d))) * t_1) * t_2;
	elseif (t_0 <= 1e+249)
		tmp = t_2 * t_1;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-71], N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.9999999999999997e-71

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. associate-*r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. associate-/l* N/A

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

   9. Applied rewrites 68.0%

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

   if -3.9999999999999997e-71 < (*.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)))) < 9.9999999999999992e248

   1. Initial program 86.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

   7. Applied rewrites 81.7%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(\frac{h}{\ell \cdot d} \cdot \left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 0.0)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (if (<= t_0 1e+249)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else if (t_0 <= 1e+249) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else if (t_0 <= 1d+249) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else if (t_0 <= 1e+249) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	elif t_0 <= 1e+249:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	elseif (t_0 <= 1e+249)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	elseif (t_0 <= 1e+249)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(N[Sqrt[N[(d / l), $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 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)))) < -0.0

   1. Initial program 76.4%

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

   3. Taylor expanded in d around inf

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

      1. *-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 16.3

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

   5. Applied rewrites 16.3%

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

   7. Applied rewrites 16.3%

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

   9. Applied rewrites 24.1%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 97.8%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 97.2%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 10^{-174}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 1e-174)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (if (<= t_0 2e+115)
       (sqrt (* (/ d l) (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 1e-174) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else if (t_0 <= 2e+115) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 1d-174) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else if (t_0 <= 2d+115) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 1e-174) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else if (t_0 <= 2e+115) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 1e-174:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	elif t_0 <= 2e+115:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 1e-174)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	elseif (t_0 <= 2e+115)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 1e-174)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	elseif (t_0 <= 2e+115)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-174], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+115], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e-174

   1. Initial program 77.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 16.7%

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

   9. Applied rewrites 24.3%

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

   if 1e-174 < (*.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)))) < 2e115

   1. Initial program 97.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 26.6%

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

   9. Applied rewrites 94.0%

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

   if 2e115 < (*.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 38.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6

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

   5. Applied rewrites 33.6%

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

   7. Applied rewrites 63.0%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-174}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 5e-163)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= t_0 2e+115)
       (sqrt (* (/ d l) (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 5e-163) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (t_0 <= 2e+115) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 5d-163) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (t_0 <= 2d+115) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 5e-163) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (t_0 <= 2e+115) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 5e-163:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif t_0 <= 2e+115:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 5e-163)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (t_0 <= 2e+115)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 5e-163)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (t_0 <= 2e+115)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-163], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+115], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999977e-163

   1. Initial program 77.3%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 17.0

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

   5. Applied rewrites 17.0%

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

   if 4.99999999999999977e-163 < (*.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)))) < 2e115

   1. Initial program 97.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 27.1%

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

   9. Applied rewrites 96.1%

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

   if 2e115 < (*.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 38.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6

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

   5. Applied rewrites 33.6%

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

   7. Applied rewrites 63.0%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\_m\right) \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      1e+249)
   (*
    (*
     (fma (* (* M_m (/ D d)) 0.5) (/ (* (* (* (/ D d) M_m) -0.25) h) l) 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+249) {
		tmp = (fma(((M_m * (D / d)) * 0.5), (((((D / d) * M_m) * -0.25) * h) / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+249)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) * 0.5), Float64(Float64(Float64(Float64(Float64(D / d) * M_m) * -0.25) * h) / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+249], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < 9.9999999999999992e248

   1. Initial program 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 85.6

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

   8. Applied rewrites 85.6%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25 \cdot t\_0, \frac{h}{\ell} \cdot \left(t\_0 \cdot 0.5\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M_m)))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        1e+249)
     (*
      (* (fma (* -0.25 t_0) (* (/ h l) (* t_0 0.5)) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (fabs (/ d (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+249) {
		tmp = (fma((-0.25 * t_0), ((h / l) * (t_0 * 0.5)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * M_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+249)
		tmp = Float64(Float64(fma(Float64(-0.25 * t_0), Float64(Float64(h / l) * Float64(t_0 * 0.5)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+249], N[(N[(N[(N[(-0.25 * t$95$0), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < 9.9999999999999992e248

   1. Initial program 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 85.6

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

   8. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. associate-/r* N/A

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

   10. Applied rewrites 85.8%

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

   if 9.9999999999999992e248 < (*.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 25.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 57.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+249}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25 \cdot \left(\frac{D}{d} \cdot M\right), \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        -5e-162)
     t_0
     (fabs t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-162)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-162)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-162)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-162], 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)))) < -5.00000000000000014e-162

   1. Initial program 82.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.6

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

   5. Applied rewrites 11.6%

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

   7. Applied rewrites 11.6%

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

   if -5.00000000000000014e-162 < (*.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 56.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

   7. Applied rewrites 65.5%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \mathsf{fma}\left(\left(M\_m \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(t\_0 \cdot -0.25\right) \cdot h}{\ell}, 1\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(-h\right)}^{-0.5} \cdot t\_2\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{t\_2}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot t\_0\right)}{\ell} \cdot M\_m, 0.5 \cdot \frac{D}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M_m))
        (t_1 (fma (* (* M_m (/ D d)) 0.5) (/ (* (* t_0 -0.25) h) l) 1.0))
        (t_2 (sqrt (- d))))
   (if (<= h -4.7e+154)
     (* (* t_1 (* (pow (- h) -0.5) t_2)) (sqrt (/ d l)))
     (if (<= h -1e-309)
       (* (* t_1 (sqrt (/ d h))) (/ t_2 (sqrt (- l))))
       (*
        (fma (* (/ (* h (* -0.25 t_0)) l) M_m) (* 0.5 (/ D d)) 1.0)
        (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double t_1 = fma(((M_m * (D / d)) * 0.5), (((t_0 * -0.25) * h) / l), 1.0);
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -4.7e+154) {
		tmp = (t_1 * (pow(-h, -0.5) * t_2)) * sqrt((d / l));
	} else if (h <= -1e-309) {
		tmp = (t_1 * sqrt((d / h))) * (t_2 / sqrt(-l));
	} else {
		tmp = fma((((h * (-0.25 * t_0)) / l) * M_m), (0.5 * (D / d)), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * M_m)
	t_1 = fma(Float64(Float64(M_m * Float64(D / d)) * 0.5), Float64(Float64(Float64(t_0 * -0.25) * h) / l), 1.0)
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -4.7e+154)
		tmp = Float64(Float64(t_1 * Float64((Float64(-h) ^ -0.5) * t_2)) * sqrt(Float64(d / l)));
	elseif (h <= -1e-309)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * Float64(t_2 / sqrt(Float64(-l))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(h * Float64(-0.25 * t_0)) / l) * M_m), Float64(0.5 * Float64(D / d)), 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(t$95$0 * -0.25), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -4.7e+154], N[(N[(t$95$1 * N[(N[Power[(-h), -0.5], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-309], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h * N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.69999999999999983e154

   1. Initial program 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 58.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 61.8

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

   8. Applied rewrites 61.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. pow-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if -4.69999999999999983e154 < h < -1.000000000000002e-309

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 81.5

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

   8. Applied rewrites 81.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 90.3

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

   10. Applied rewrites 90.3%

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

   if -1.000000000000002e-309 < h

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 57.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 62.5

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 74.2%

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

Alternative 12: 78.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \sqrt{-d}\\ t_2 := \mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot t\_0\right)}{\ell} \cdot M\_m, 0.5 \cdot \frac{D}{d}, 1\right)\\ \mathbf{if}\;h \leq -4 \cdot 10^{+209}:\\ \;\;\;\;\frac{t\_2 \cdot t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(t\_0 \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{t\_1}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M_m))
        (t_1 (sqrt (- d)))
        (t_2 (fma (* (/ (* h (* -0.25 t_0)) l) M_m) (* 0.5 (/ D d)) 1.0)))
   (if (<= h -4e+209)
     (* (/ (* t_2 t_1) (sqrt (- h))) (sqrt (/ d l)))
     (if (<= h -1e-309)
       (*
        (*
         (fma (* (* M_m (/ D d)) 0.5) (/ (* (* t_0 -0.25) h) l) 1.0)
         (sqrt (/ d h)))
        (/ t_1 (sqrt (- l))))
       (* t_2 (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double t_1 = sqrt(-d);
	double t_2 = fma((((h * (-0.25 * t_0)) / l) * M_m), (0.5 * (D / d)), 1.0);
	double tmp;
	if (h <= -4e+209) {
		tmp = ((t_2 * t_1) / sqrt(-h)) * sqrt((d / l));
	} else if (h <= -1e-309) {
		tmp = (fma(((M_m * (D / d)) * 0.5), (((t_0 * -0.25) * h) / l), 1.0) * sqrt((d / h))) * (t_1 / sqrt(-l));
	} else {
		tmp = t_2 * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * M_m)
	t_1 = sqrt(Float64(-d))
	t_2 = fma(Float64(Float64(Float64(h * Float64(-0.25 * t_0)) / l) * M_m), Float64(0.5 * Float64(D / d)), 1.0)
	tmp = 0.0
	if (h <= -4e+209)
		tmp = Float64(Float64(Float64(t_2 * t_1) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (h <= -1e-309)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) * 0.5), Float64(Float64(Float64(t_0 * -0.25) * h) / l), 1.0) * sqrt(Float64(d / h))) * Float64(t_1 / sqrt(Float64(-l))));
	else
		tmp = Float64(t_2 * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(h * N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -4e+209], N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-309], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(t$95$0 * -0.25), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.0000000000000003e209

   1. Initial program 48.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 43.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 48.6

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

   8. Applied rewrites 48.6%

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

   10. Applied rewrites 78.2%

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

   if -4.0000000000000003e209 < h < -1.000000000000002e-309

   1. Initial program 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 82.8

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

   8. Applied rewrites 82.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 91.0

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

   10. Applied rewrites 91.0%

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

   if -1.000000000000002e-309 < h

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 57.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 62.5

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 74.2%

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

Alternative 13: 78.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot t\_0\right)}{\ell} \cdot M\_m, 0.5 \cdot \frac{D}{d}, 1\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(t\_0 \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot t\_2\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 10^{+117}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_2 \cdot t\_1\right) \cdot \sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M_m))
        (t_1 (fma (* (/ (* h (* -0.25 t_0)) l) M_m) (* 0.5 (/ D d)) 1.0))
        (t_2 (sqrt (/ d h))))
   (if (<= l -5e-310)
     (*
      (* (fma (* (* M_m (/ D d)) 0.5) (/ (* (* t_0 -0.25) h) l) 1.0) t_2)
      (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l 1e+117)
       (* t_1 (/ d (sqrt (* l h))))
       (/ (* (* t_2 t_1) (sqrt d)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double t_1 = fma((((h * (-0.25 * t_0)) / l) * M_m), (0.5 * (D / d)), 1.0);
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= -5e-310) {
		tmp = (fma(((M_m * (D / d)) * 0.5), (((t_0 * -0.25) * h) / l), 1.0) * t_2) * (sqrt(-d) / sqrt(-l));
	} else if (l <= 1e+117) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = ((t_2 * t_1) * sqrt(d)) / sqrt(l);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * M_m)
	t_1 = fma(Float64(Float64(Float64(h * Float64(-0.25 * t_0)) / l) * M_m), Float64(0.5 * Float64(D / d)), 1.0)
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) * 0.5), Float64(Float64(Float64(t_0 * -0.25) * h) / l), 1.0) * t_2) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= 1e+117)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(t_2 * t_1) * sqrt(d)) / sqrt(l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h * N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(t$95$0 * -0.25), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+117], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. pow-pow N/A

         \[\leadsto \left(\left(\color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. inv-pow N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. frac-times N/A

          \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. clear-num N/A

          \[\leadsto \left(\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 75.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)} \cdot \frac{-1}{2}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. metadata-eval 77.0

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{-0.25}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 77.0%

      \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. frac-2neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(\ell\right)}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{-1}{4}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

      9. lower-neg.f64 83.9

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}} \]

   10. Applied rewrites 83.9%

       \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \]

   if -4.999999999999985e-310 < l < 1.00000000000000005e117

   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-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. pow-pow N/A

         \[\leadsto \left(\left(\color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. inv-pow N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. frac-times N/A

          \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. clear-num N/A

          \[\leadsto \left(\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 64.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)} \cdot \frac{-1}{2}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. metadata-eval 67.2

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{-0.25}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 67.2%

      \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 82.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot \left(\frac{D}{d} \cdot M\right)\right)}{\ell} \cdot M, 0.5 \cdot \frac{D}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 1.00000000000000005e117 < l

   1. Initial program 48.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. pow-pow N/A

         \[\leadsto \left(\left(\color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. inv-pow N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. frac-times N/A

          \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. clear-num N/A

          \[\leadsto \left(\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 50.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)} \cdot \frac{-1}{2}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. metadata-eval 49.1

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{-0.25}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 49.1%

      \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 68.1%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot \left(\frac{D}{d} \cdot M\right)\right)}{\ell} \cdot M, 0.5 \cdot \frac{D}{d}, 1\right)\right) \cdot \sqrt{d}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-297}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot \left(\frac{D}{d} \cdot M\_m\right)\right)}{\ell} \cdot M\_m, 0.5 \cdot \frac{D}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.8e-107)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 2e-297)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (*
      (fma (* (/ (* h (* -0.25 (* (/ D d) M_m))) l) M_m) (* 0.5 (/ D d)) 1.0)
      (/ d (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.8e-107) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 2e-297) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else {
		tmp = fma((((h * (-0.25 * ((D / d) * M_m))) / l) * M_m), (0.5 * (D / d)), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.8e-107)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 2e-297)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(h * Float64(-0.25 * Float64(Float64(D / d) * M_m))) / l) * M_m), Float64(0.5 * Float64(D / d)), 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.8e-107], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e-297], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h * N[(-0.25 * N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.79999999999999988e-107

   1. Initial program 78.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-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 6.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 6.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   if -1.79999999999999988e-107 < d < 2.00000000000000008e-297

   1. Initial program 59.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 22.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 22.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.8%

      \[\leadsto \frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}} \]

   if 2.00000000000000008e-297 < d

   1. Initial program 60.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. pow-pow N/A

         \[\leadsto \left(\left(\color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. inv-pow N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. frac-times N/A

          \[\leadsto \left(\left({\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. clear-num N/A

          \[\leadsto \left(\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 62.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{2}\right) \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}\right)} \cdot \frac{-1}{2}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{1}{2}, \frac{\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(\frac{1}{2} \cdot \frac{-1}{2}\right)\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. metadata-eval 63.7

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{-0.25}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 63.7%

      \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5, \color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.25\right) \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 75.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(-0.25 \cdot \left(\frac{D}{d} \cdot M\right)\right)}{\ell} \cdot M, 0.5 \cdot \frac{D}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$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 d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 24.5

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(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)))))