Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 83.0%

Time: 17.2s

Alternatives: 19

Speedup: 1.0×

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: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M\_m}{d}}{{h}^{-1}}\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\_m\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M\_m} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \end{array} \end{array} \]

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 l) (pow 2.0 -1.0))))
   (if (<= d -2.4e-169)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
      (-
       1.0
       (*
        (/ (* (* (/ 0.5 d) M_m) D) l)
        (/ (* (* 0.5 (* D 0.5)) (/ M_m d)) (pow h -1.0)))))
     (if (<= d 5.9e-114)
       (/
        (* (sqrt (/ h l)) (fma (* -0.125 (/ (pow (* D M_m) 2.0) d)) (/ h l) d))
        h)
       (*
        (* (/ (sqrt d) (sqrt h)) t_0)
        (fma (/ (* -0.5 (pow (* (/ (/ d D) M_m) 2.0) -2.0)) l) h 1.0))))))

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 = pow((d / l), pow(2.0, -1.0));
	double tmp;
	if (d <= -2.4e-169) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (((((0.5 / d) * M_m) * D) / l) * (((0.5 * (D * 0.5)) * (M_m / d)) / pow(h, -1.0))));
	} else if (d <= 5.9e-114) {
		tmp = (sqrt((h / l)) * fma((-0.125 * (pow((D * M_m), 2.0) / d)), (h / l), d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * fma(((-0.5 * pow((((d / D) / M_m) * 2.0), -2.0)) / l), h, 1.0);
	}
	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 / l) ^ (2.0 ^ -1.0)
	tmp = 0.0
	if (d <= -2.4e-169)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 / d) * M_m) * D) / l) * Float64(Float64(Float64(0.5 * Float64(D * 0.5)) * Float64(M_m / d)) / (h ^ -1.0)))));
	elseif (d <= 5.9e-114)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(-0.125 * Float64((Float64(D * M_m) ^ 2.0) / d)), Float64(h / l), d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * fma(Float64(Float64(-0.5 * (Float64(Float64(Float64(d / D) / M_m) * 2.0) ^ -2.0)) / l), h, 1.0));
	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[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.4e-169], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(0.5 * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.9e-114], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(-0.5 * N[Power[N[(N[(N[(d / D), $MachinePrecision] / M$95$m), $MachinePrecision] * 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.40000000000000011e-169

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 89.3

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

   6. Applied rewrites 89.3%

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

   if -2.40000000000000011e-169 < d < 5.9000000000000001e-114

   1. Initial program 59.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 35.2%

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

   10. Applied rewrites 61.1%

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

   12. Applied rewrites 82.2%

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

   if 5.9000000000000001e-114 < d

   1. Initial program 72.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 94.6

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

   6. Applied rewrites 94.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\ell \cdot h}\\ t_3 := \mathsf{fma}\left(t\_0, t\_0 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{t\_2}}{\ell}\\ t_4 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-h}{\ell} \cdot \left(\frac{M\_m}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right), D \cdot t\_4, t\_4\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M\_m}\right)}^{-2} \cdot -0.125, h, \ell\right) \cdot \left|d\right|}{t\_2 \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (* (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_2 (sqrt (* l h)))
        (t_3 (* (fma t_0 (* t_0 (* -0.125 h)) l) (/ (/ (fabs d) t_2) l)))
        (t_4 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1e-110)
       (fma
        (* (* (/ (- h) l) (* (/ M_m d) (* 0.25 D))) (* (/ 0.5 d) M_m))
        (* D t_4)
        t_4)
       (if (<= t_1 0.0)
         (/
          (* (fma (* (pow (/ (/ d D) M_m) -2.0) -0.125) h l) (fabs d))
          (* t_2 l))
         (if (<= t_1 4e+282)
           (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
           t_3))))))

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 = (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_2 = sqrt((l * h));
	double t_3 = fma(t_0, (t_0 * (-0.125 * h)), l) * ((fabs(d) / t_2) / l);
	double t_4 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1e-110) {
		tmp = fma((((-h / l) * ((M_m / d) * (0.25 * D))) * ((0.5 / d) * M_m)), (D * t_4), t_4);
	} else if (t_1 <= 0.0) {
		tmp = (fma((pow(((d / D) / M_m), -2.0) * -0.125), h, l) * fabs(d)) / (t_2 * l);
	} else if (t_1 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_3;
	}
	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 = 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_2 = sqrt(Float64(l * h))
	t_3 = Float64(fma(t_0, Float64(t_0 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / t_2) / l))
	t_4 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1e-110)
		tmp = fma(Float64(Float64(Float64(Float64(-h) / l) * Float64(Float64(M_m / d) * Float64(0.25 * D))) * Float64(Float64(0.5 / d) * M_m)), Float64(D * t_4), t_4);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64((Float64(Float64(d / D) / M_m) ^ -2.0) * -0.125), h, l) * abs(d)) / Float64(t_2 * l));
	elseif (t_1 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_3;
	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[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$2 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$0 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1e-110], N[(N[(N[(N[((-h) / l), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D * t$95$4), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[Power[N[(N[(d / D), $MachinePrecision] / M$95$m), $MachinePrecision], -2.0], $MachinePrecision] * -0.125), $MachinePrecision] * h + l), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -inf.0 or 4.00000000000000013e282 < (*.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 49.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 59.2%

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

   6. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 40.9%

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

   10. Applied rewrites 81.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-110

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 83.3%

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

   6. Applied rewrites 72.3%

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

   if -1.0000000000000001e-110 < (*.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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 32.3%

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

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 76.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\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 -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), D \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{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 0:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125, h, \ell\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h} \cdot \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 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\ell \cdot h}\\ t_3 := \mathsf{fma}\left(t\_0, t\_0 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{t\_2}}{\ell}\\ t_4 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-h}{\ell} \cdot \left(\frac{M\_m}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right), D \cdot t\_4, t\_4\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_2} \cdot 1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (* (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_2 (sqrt (* l h)))
        (t_3 (* (fma t_0 (* t_0 (* -0.125 h)) l) (/ (/ (fabs d) t_2) l)))
        (t_4 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1e-110)
       (fma
        (* (* (/ (- h) l) (* (/ M_m d) (* 0.25 D))) (* (/ 0.5 d) M_m))
        (* D t_4)
        t_4)
       (if (<= t_1 0.0)
         (* (/ (sqrt (* d d)) t_2) 1.0)
         (if (<= t_1 4e+282)
           (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
           t_3))))))

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 = (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_2 = sqrt((l * h));
	double t_3 = fma(t_0, (t_0 * (-0.125 * h)), l) * ((fabs(d) / t_2) / l);
	double t_4 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1e-110) {
		tmp = fma((((-h / l) * ((M_m / d) * (0.25 * D))) * ((0.5 / d) * M_m)), (D * t_4), t_4);
	} else if (t_1 <= 0.0) {
		tmp = (sqrt((d * d)) / t_2) * 1.0;
	} else if (t_1 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_3;
	}
	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 = 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_2 = sqrt(Float64(l * h))
	t_3 = Float64(fma(t_0, Float64(t_0 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / t_2) / l))
	t_4 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1e-110)
		tmp = fma(Float64(Float64(Float64(Float64(-h) / l) * Float64(Float64(M_m / d) * Float64(0.25 * D))) * Float64(Float64(0.5 / d) * M_m)), Float64(D * t_4), t_4);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(d * d)) / t_2) * 1.0);
	elseif (t_1 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_3;
	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[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$2 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$0 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1e-110], N[(N[(N[(N[((-h) / l), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D * t$95$4), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -inf.0 or 4.00000000000000013e282 < (*.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 49.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 59.2%

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

   6. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 40.9%

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

   10. Applied rewrites 81.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-110

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 83.3%

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

   6. Applied rewrites 72.3%

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

   if -1.0000000000000001e-110 < (*.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 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

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 38.2

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

   7. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), D \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\ell \cdot h}\\ t_3 := \mathsf{fma}\left(t\_0, t\_0 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{t\_2}}{\ell}\\ t_4 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D, \left(\frac{-h}{\ell} \cdot \left(\frac{M\_m}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot t\_4, t\_4\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_2} \cdot 1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (* (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_2 (sqrt (* l h)))
        (t_3 (* (fma t_0 (* t_0 (* -0.125 h)) l) (/ (/ (fabs d) t_2) l)))
        (t_4 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1e-110)
       (fma
        (* (* (/ 0.5 d) M_m) D)
        (* (* (/ (- h) l) (* (/ M_m d) (* 0.25 D))) t_4)
        t_4)
       (if (<= t_1 0.0)
         (* (/ (sqrt (* d d)) t_2) 1.0)
         (if (<= t_1 4e+282)
           (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
           t_3))))))

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 = (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_2 = sqrt((l * h));
	double t_3 = fma(t_0, (t_0 * (-0.125 * h)), l) * ((fabs(d) / t_2) / l);
	double t_4 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1e-110) {
		tmp = fma((((0.5 / d) * M_m) * D), (((-h / l) * ((M_m / d) * (0.25 * D))) * t_4), t_4);
	} else if (t_1 <= 0.0) {
		tmp = (sqrt((d * d)) / t_2) * 1.0;
	} else if (t_1 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_3;
	}
	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 = 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_2 = sqrt(Float64(l * h))
	t_3 = Float64(fma(t_0, Float64(t_0 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / t_2) / l))
	t_4 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1e-110)
		tmp = fma(Float64(Float64(Float64(0.5 / d) * M_m) * D), Float64(Float64(Float64(Float64(-h) / l) * Float64(Float64(M_m / d) * Float64(0.25 * D))) * t_4), t_4);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(d * d)) / t_2) * 1.0);
	elseif (t_1 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_3;
	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[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$2 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$0 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1e-110], N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] * N[(N[(N[((-h) / l), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -inf.0 or 4.00000000000000013e282 < (*.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 49.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 59.2%

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

   6. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 40.9%

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

   10. Applied rewrites 81.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-110

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 83.3%

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

   6. Applied rewrites 93.5%

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

   if -1.0000000000000001e-110 < (*.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 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

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 38.2

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

   7. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{-h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := \frac{D}{d} \cdot M\_m\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \sqrt{\ell \cdot h}\\ t_4 := \mathsf{fma}\left(t\_1, t\_1 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{t\_3}}{\ell}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(\left(\frac{-h}{\ell} \cdot \left(\frac{M\_m}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right)\right), D, t\_0\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_3} \cdot 1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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 (sqrt (* (/ d l) (/ d h))))
        (t_1 (* (/ D d) M_m))
        (t_2
         (*
          (* (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_3 (sqrt (* l h)))
        (t_4 (* (fma t_1 (* t_1 (* -0.125 h)) l) (/ (/ (fabs d) t_3) l))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 -1e-110)
       (fma
        (* t_0 (* (* (/ (- h) l) (* (/ M_m d) (* 0.25 D))) (* (/ 0.5 d) M_m)))
        D
        t_0)
       (if (<= t_2 0.0)
         (* (/ (sqrt (* d d)) t_3) 1.0)
         (if (<= t_2 4e+282)
           (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
           t_4))))))

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 = sqrt(((d / l) * (d / h)));
	double t_1 = (D / d) * M_m;
	double t_2 = (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_3 = sqrt((l * h));
	double t_4 = fma(t_1, (t_1 * (-0.125 * h)), l) * ((fabs(d) / t_3) / l);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= -1e-110) {
		tmp = fma((t_0 * (((-h / l) * ((M_m / d) * (0.25 * D))) * ((0.5 / d) * M_m))), D, t_0);
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((d * d)) / t_3) * 1.0;
	} else if (t_2 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_4;
	}
	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 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = Float64(Float64(D / d) * M_m)
	t_2 = 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_3 = sqrt(Float64(l * h))
	t_4 = Float64(fma(t_1, Float64(t_1 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / t_3) / l))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= -1e-110)
		tmp = fma(Float64(t_0 * Float64(Float64(Float64(Float64(-h) / l) * Float64(Float64(M_m / d) * Float64(0.25 * D))) * Float64(Float64(0.5 / d) * M_m))), D, t_0);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(d * d)) / t_3) * 1.0);
	elseif (t_2 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_4;
	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[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * N[(t$95$1 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$3), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, -1e-110], N[(N[(t$95$0 * N[(N[(N[((-h) / l), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -inf.0 or 4.00000000000000013e282 < (*.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 49.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 59.2%

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

   6. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 40.9%

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

   10. Applied rewrites 81.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-110

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 83.3%

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

   6. Applied rewrites 72.7%

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

   if -1.0000000000000001e-110 < (*.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 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

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 38.2

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

   7. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(\left(\frac{-h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right), D, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \frac{\sqrt{d \cdot d}}{t\_1} \cdot 1\\ t_3 := \left(\left(-0.125 \cdot \left(h \cdot \frac{D \cdot D}{d}\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \frac{\frac{\left|d\right|}{t\_1}}{\ell}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (* l h)))
        (t_2 (* (/ (sqrt (* d d)) t_1) 1.0))
        (t_3
         (*
          (* (* -0.125 (* h (/ (* D D) d))) (/ (* M_m M_m) d))
          (/ (/ (fabs d) t_1) l))))
   (if (<= t_0 -500.0)
     t_3
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 4e+282)
         (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 t_3))))))

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((l * h));
	double t_2 = (sqrt((d * d)) / t_1) * 1.0;
	double t_3 = ((-0.125 * (h * ((D * D) / d))) * ((M_m * M_m) / d)) * ((fabs(d) / t_1) / l);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_3;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

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((l * h));
	double t_2 = (Math.sqrt((d * d)) / t_1) * 1.0;
	double t_3 = ((-0.125 * (h * ((D * D) / d))) * ((M_m * M_m) / d)) * ((Math.abs(d) / t_1) / l);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_3;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 4e+282) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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((l * h))
	t_2 = (math.sqrt((d * d)) / t_1) * 1.0
	t_3 = ((-0.125 * (h * ((D * D) / d))) * ((M_m * M_m) / d)) * ((math.fabs(d) / t_1) / l)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_3
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 4e+282:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = t_3
	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(l * h))
	t_2 = Float64(Float64(sqrt(Float64(d * d)) / t_1) * 1.0)
	t_3 = Float64(Float64(Float64(-0.125 * Float64(h * Float64(Float64(D * D) / d))) * Float64(Float64(M_m * M_m) / d)) * Float64(Float64(abs(d) / t_1) / l))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_3;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	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((l * h));
	t_2 = (sqrt((d * d)) / t_1) * 1.0;
	t_3 = ((-0.125 * (h * ((D * D) / d))) * ((M_m * M_m) / d)) * ((abs(d) / t_1) / l);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_3;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 4e+282)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	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[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.125 * N[(h * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$3, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -500 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 66.4%

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

   6. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 59.0

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

   11. Applied rewrites 59.0%

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

   if -500 < (*.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 or 4.00000000000000013e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 45.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 35.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 35.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 36.8

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

   7. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 58.1

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

   9. Applied rewrites 58.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -500:\\ \;\;\;\;\left(\left(-0.125 \cdot \left(h \cdot \frac{D \cdot D}{d}\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.125 \cdot \left(h \cdot \frac{D \cdot D}{d}\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\ell \cdot h}\\ t_3 := \mathsf{fma}\left(t\_0, t\_0 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{t\_2}}{\ell}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_2} \cdot 1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (* (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_2 (sqrt (* l h)))
        (t_3 (* (fma t_0 (* t_0 (* -0.125 h)) l) (/ (/ (fabs d) t_2) l))))
   (if (<= t_1 -1e-110)
     t_3
     (if (<= t_1 0.0)
       (* (/ (sqrt (* d d)) t_2) 1.0)
       (if (<= t_1 4e+282) (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h))) t_3)))))

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 = (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_2 = sqrt((l * h));
	double t_3 = fma(t_0, (t_0 * (-0.125 * h)), l) * ((fabs(d) / t_2) / l);
	double tmp;
	if (t_1 <= -1e-110) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = (sqrt((d * d)) / t_2) * 1.0;
	} else if (t_1 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_3;
	}
	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 = 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_2 = sqrt(Float64(l * h))
	t_3 = Float64(fma(t_0, Float64(t_0 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / t_2) / l))
	tmp = 0.0
	if (t_1 <= -1e-110)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(d * d)) / t_2) * 1.0);
	elseif (t_1 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_3;
	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[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$2 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$0 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-110], t$95$3, If[LessEqual[t$95$1, 0.0], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-110 or 4.00000000000000013e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 54.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 61.9%

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

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 40.2%

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

   10. Applied rewrites 78.8%

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

   if -1.0000000000000001e-110 < (*.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 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

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 38.2

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

   7. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\_m\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \end{array} \]

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
         (*
          (* (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_1 2e-183)
     (/
      (* (sqrt (/ h l)) (fma (* -0.125 (/ (pow (* D M_m) 2.0) d)) (/ h l) d))
      h)
     (if (<= t_1 4e+282)
       (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
       (*
        (fma t_0 (* t_0 (* -0.125 h)) l)
        (/ (/ (fabs d) (sqrt (* l h))) 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 = (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_1 <= 2e-183) {
		tmp = (sqrt((h / l)) * fma((-0.125 * (pow((D * M_m), 2.0) / d)), (h / l), d)) / h;
	} else if (t_1 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fma(t_0, (t_0 * (-0.125 * h)), l) * ((fabs(d) / sqrt((l * h))) / 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 = 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_1 <= 2e-183)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(-0.125 * Float64((Float64(D * M_m) ^ 2.0) / d)), Float64(h / l), d)) / h);
	elseif (t_1 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(t_0, Float64(t_0 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / sqrt(Float64(l * h))) / 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[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$1, 2e-183], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$0 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $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)))) < 2.00000000000000001e-183

   1. Initial program 80.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 35.6%

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

   10. Applied rewrites 58.0%

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

   12. Applied rewrites 81.7%

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

   if 2.00000000000000001e-183 < (*.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)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   if 4.00000000000000013e282 < (*.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 17.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 31.5%

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

   6. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 73.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \end{array} \]

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 D) -0.125) (* (/ (* M_m M_m) d) (/ (sqrt (/ h l)) (fabs l))))
     (if (<= t_0 4e+282)
       (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
       (* (/ (sqrt (* d d)) (sqrt (* l h))) 1.0)))))

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 = (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 * D) * -0.125) * (((M_m * M_m) / d) * (sqrt((h / l)) / fabs(l)));
	} else if (t_0 <= 4e+282) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = (sqrt((d * d)) / sqrt((l * h))) * 1.0;
	}
	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 / 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_1 * d_1) * (-0.125d0)) * (((m_m * m_m) / d) * (sqrt((h / l)) / abs(l)))
    else if (t_0 <= 4d+282) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = (sqrt((d * d)) / sqrt((l * h))) * 1.0d0
    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 * D) * -0.125) * (((M_m * M_m) / d) * (Math.sqrt((h / l)) / Math.abs(l)));
	} else if (t_0 <= 4e+282) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = (Math.sqrt((d * d)) / Math.sqrt((l * h))) * 1.0;
	}
	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 = (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 * D) * -0.125) * (((M_m * M_m) / d) * (math.sqrt((h / l)) / math.fabs(l)))
	elif t_0 <= 4e+282:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = (math.sqrt((d * d)) / math.sqrt((l * h))) * 1.0
	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(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(Float64(Float64(D * D) * -0.125) * Float64(Float64(Float64(M_m * M_m) / d) * Float64(sqrt(Float64(h / l)) / abs(l))));
	elseif (t_0 <= 4e+282)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(sqrt(Float64(d * d)) / sqrt(Float64(l * h))) * 1.0);
	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 / 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 * D) * -0.125) * (((M_m * M_m) / d) * (sqrt((h / l)) / abs(l)));
	elseif (t_0 <= 4e+282)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = (sqrt((d * d)) / sqrt((l * h))) * 1.0;
	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[(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[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+282], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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 79.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 37.7

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

   8. Applied rewrites 37.7%

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

   10. Applied rewrites 41.5%

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

   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)))) < 4.00000000000000013e282

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   if 4.00000000000000013e282 < (*.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 17.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 26.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 26.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 37.7

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

   9. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M \cdot M}{d} \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := \frac{D}{d} \cdot M\_m\\ t_2 := {\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)}\\ \mathbf{if}\;\left(t\_2 \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot t\_0 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \end{array} \]

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
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (* (/ D d) M_m))
        (t_2 (pow (/ d h) (pow 2.0 -1.0))))
   (if (<= (* (* t_2 (pow (/ d l) (pow 2.0 -1.0))) t_0) 4e+282)
     (* (* t_2 (sqrt (/ d l))) t_0)
     (* (fma t_1 (* t_1 (* -0.125 h)) l) (/ (/ (fabs d) (sqrt (* l h))) 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 = 1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = (D / d) * M_m;
	double t_2 = pow((d / h), pow(2.0, -1.0));
	double tmp;
	if (((t_2 * pow((d / l), pow(2.0, -1.0))) * t_0) <= 4e+282) {
		tmp = (t_2 * sqrt((d / l))) * t_0;
	} else {
		tmp = fma(t_1, (t_1 * (-0.125 * h)), l) * ((fabs(d) / sqrt((l * h))) / 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(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(Float64(D / d) * M_m)
	t_2 = Float64(d / h) ^ (2.0 ^ -1.0)
	tmp = 0.0
	if (Float64(Float64(t_2 * (Float64(d / l) ^ (2.0 ^ -1.0))) * t_0) <= 4e+282)
		tmp = Float64(Float64(t_2 * sqrt(Float64(d / l))) * t_0);
	else
		tmp = Float64(fma(t_1, Float64(t_1 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / sqrt(Float64(l * h))) / 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[(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]}, Block[{t$95$1 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 4e+282], N[(N[(t$95$2 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$1 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 88.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 88.1

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

   4. Applied rewrites 88.1%

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

   if 4.00000000000000013e282 < (*.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 17.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 31.5%

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

   6. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 73.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)}\\ t_1 := \frac{D}{d} \cdot M\_m\\ \mathbf{if}\;\left(t\_0 \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M\_m} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1 \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \end{array} \]

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 (pow (/ d h) (pow 2.0 -1.0))) (t_1 (* (/ D d) M_m)))
   (if (<=
        (*
         (* t_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))))
        4e+282)
     (*
      (* t_0 (sqrt (/ d l)))
      (fma (/ (* -0.5 (pow (* (/ (/ d D) M_m) 2.0) -2.0)) l) h 1.0))
     (* (fma t_1 (* t_1 (* -0.125 h)) l) (/ (/ (fabs d) (sqrt (* l h))) 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 = pow((d / h), pow(2.0, -1.0));
	double t_1 = (D / d) * M_m;
	double tmp;
	if (((t_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)))) <= 4e+282) {
		tmp = (t_0 * sqrt((d / l))) * fma(((-0.5 * pow((((d / D) / M_m) * 2.0), -2.0)) / l), h, 1.0);
	} else {
		tmp = fma(t_1, (t_1 * (-0.125 * h)), l) * ((fabs(d) / sqrt((l * h))) / 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 / h) ^ (2.0 ^ -1.0)
	t_1 = Float64(Float64(D / d) * M_m)
	tmp = 0.0
	if (Float64(Float64(t_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)))) <= 4e+282)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * fma(Float64(Float64(-0.5 * (Float64(Float64(Float64(d / D) / M_m) * 2.0) ^ -2.0)) / l), h, 1.0));
	else
		tmp = Float64(fma(t_1, Float64(t_1 * Float64(-0.125 * h)), l) * Float64(Float64(abs(d) / sqrt(Float64(l * h))) / 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[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * 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], 4e+282], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[Power[N[(N[(N[(d / D), $MachinePrecision] / M$95$m), $MachinePrecision] * 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$1 * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000013e282

   1. Initial program 88.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 87.5

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

   6. Applied rewrites 87.5%

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

   if 4.00000000000000013e282 < (*.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 17.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 31.5%

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

   6. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125\right) \cdot h\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}, \ell \cdot 1\right)}{\ell \cdot \sqrt{\frac{\frac{h}{d}}{\frac{d}{\ell}}}}} \]

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 73.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d} \cdot M, \left(\frac{D}{d} \cdot M\right) \cdot \left(-0.125 \cdot h\right), \ell\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\\ t_1 := \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M\_m} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{-168}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\_m\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

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 l) (pow 2.0 -1.0)))
        (t_1 (fma (/ (* -0.5 (pow (* (/ (/ d D) M_m) 2.0) -2.0)) l) h 1.0)))
   (if (<= d -5.8e-168)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_1)
     (if (<= d 5.9e-114)
       (/
        (* (sqrt (/ h l)) (fma (* -0.125 (/ (pow (* D M_m) 2.0) d)) (/ h l) d))
        h)
       (* (* (/ (sqrt d) (sqrt h)) t_0) t_1)))))

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 / l), pow(2.0, -1.0));
	double t_1 = fma(((-0.5 * pow((((d / D) / M_m) * 2.0), -2.0)) / l), h, 1.0);
	double tmp;
	if (d <= -5.8e-168) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1;
	} else if (d <= 5.9e-114) {
		tmp = (sqrt((h / l)) * fma((-0.125 * (pow((D * M_m), 2.0) / d)), (h / l), d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_1;
	}
	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 / l) ^ (2.0 ^ -1.0)
	t_1 = fma(Float64(Float64(-0.5 * (Float64(Float64(Float64(d / D) / M_m) * 2.0) ^ -2.0)) / l), h, 1.0)
	tmp = 0.0
	if (d <= -5.8e-168)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_1);
	elseif (d <= 5.9e-114)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(-0.125 * Float64((Float64(D * M_m) ^ 2.0) / d)), Float64(h / l), d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * t_1);
	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[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.5 * N[Power[N[(N[(N[(d / D), $MachinePrecision] / M$95$m), $MachinePrecision] * 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]}, If[LessEqual[d, -5.8e-168], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 5.9e-114], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.7999999999999997e-168

   1. Initial program 69.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 88.7

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

   6. Applied rewrites 88.7%

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

   if -5.7999999999999997e-168 < d < 5.9000000000000001e-114

   1. Initial program 59.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 35.2%

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

   10. Applied rewrites 61.1%

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

   12. Applied rewrites 82.2%

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

   if 5.9000000000000001e-114 < d

   1. Initial program 72.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 94.6

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

   6. Applied rewrites 94.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-168}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M\_m} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{-54}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\_m\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot t\_0\\ \end{array} \end{array} \]

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 (fma (/ (* -0.5 (pow (* (/ (/ d D) M_m) 2.0) -2.0)) l) h 1.0)))
   (if (<= d -9.6e-54)
     (* (* (pow (/ d h) (pow 2.0 -1.0)) (sqrt (/ d l))) t_0)
     (if (<= d 5.9e-114)
       (/
        (* (sqrt (/ h l)) (fma (* -0.125 (/ (pow (* D M_m) 2.0) d)) (/ h l) d))
        h)
       (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0))) 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 = fma(((-0.5 * pow((((d / D) / M_m) * 2.0), -2.0)) / l), h, 1.0);
	double tmp;
	if (d <= -9.6e-54) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * sqrt((d / l))) * t_0;
	} else if (d <= 5.9e-114) {
		tmp = (sqrt((h / l)) * fma((-0.125 * (pow((D * M_m), 2.0) / d)), (h / l), d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * 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 = fma(Float64(Float64(-0.5 * (Float64(Float64(Float64(d / D) / M_m) * 2.0) ^ -2.0)) / l), h, 1.0)
	tmp = 0.0
	if (d <= -9.6e-54)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * sqrt(Float64(d / l))) * t_0);
	elseif (d <= 5.9e-114)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(-0.125 * Float64((Float64(D * M_m) ^ 2.0) / d)), Float64(h / l), d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * t_0);
	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[(-0.5 * N[Power[N[(N[(N[(d / D), $MachinePrecision] / M$95$m), $MachinePrecision] * 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]}, If[LessEqual[d, -9.6e-54], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 5.9e-114], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.60000000000000053e-54

   1. Initial program 76.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 83.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 83.8

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

   6. Applied rewrites 83.8%

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

   if -9.60000000000000053e-54 < d < 5.9000000000000001e-114

   1. Initial program 56.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 34.2%

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

   10. Applied rewrites 58.9%

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

   12. Applied rewrites 78.7%

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

   if 5.9000000000000001e-114 < d

   1. Initial program 72.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 94.6

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

   6. Applied rewrites 94.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-54}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \frac{h}{\ell}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-136}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{\frac{-d}{h} \cdot d}}{\sqrt{-\ell}} \cdot 1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= l -3e-136)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l -2.55e-306)
     (* (/ (sqrt (* (/ (- d) h) d)) (sqrt (- l))) 1.0)
     (if (<= l 5e-236)
       (/ (* (- d) (sqrt (/ h l))) h)
       (/ d (* (sqrt l) (sqrt 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 (l <= -3e-136) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= -2.55e-306) {
		tmp = (sqrt(((-d / h) * d)) / sqrt(-l)) * 1.0;
	} else if (l <= 5e-236) {
		tmp = (-d * sqrt((h / l))) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: tmp
    if (l <= (-3d-136)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (l <= (-2.55d-306)) then
        tmp = (sqrt(((-d / h) * d)) / sqrt(-l)) * 1.0d0
    else if (l <= 5d-236) then
        tmp = (-d * sqrt((h / l))) / h
    else
        tmp = d / (sqrt(l) * sqrt(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 tmp;
	if (l <= -3e-136) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (l <= -2.55e-306) {
		tmp = (Math.sqrt(((-d / h) * d)) / Math.sqrt(-l)) * 1.0;
	} else if (l <= 5e-236) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	tmp = 0
	if l <= -3e-136:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif l <= -2.55e-306:
		tmp = (math.sqrt(((-d / h) * d)) / math.sqrt(-l)) * 1.0
	elif l <= 5e-236:
		tmp = (-d * math.sqrt((h / l))) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 (l <= -3e-136)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= -2.55e-306)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(-d) / h) * d)) / sqrt(Float64(-l))) * 1.0);
	elseif (l <= 5e-236)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (l <= -3e-136)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (l <= -2.55e-306)
		tmp = (sqrt(((-d / h) * d)) / sqrt(-l)) * 1.0;
	elseif (l <= 5e-236)
		tmp = (-d * sqrt((h / l))) / h;
	else
		tmp = d / (sqrt(l) * sqrt(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_] := If[LessEqual[l, -3e-136], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.55e-306], N[(N[(N[Sqrt[N[(N[((-d) / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 5e-236], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.9999999999999998e-136

   1. Initial program 65.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 52.6

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

   8. Applied rewrites 52.6%

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

   if -2.9999999999999998e-136 < l < -2.54999999999999986e-306

   1. Initial program 66.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 40.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 40.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 43.5

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

   7. Applied rewrites 43.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. sqrt-unprod N/A

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

      13. frac-2neg N/A

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

      14. associate-*r/ N/A

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

      15. sqrt-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-neg.f64 46.2

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

   9. Applied rewrites 46.2%

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

   if -2.54999999999999986e-306 < l < 4.9999999999999998e-236

   1. Initial program 93.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 13.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in l around -inf

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

   11. Applied rewrites 86.7%

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

   if 4.9999999999999998e-236 < l

   1. Initial program 66.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.4

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

   8. Applied rewrites 49.4%

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

   10. Applied rewrites 49.9%

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

   12. Applied rewrites 56.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-136}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{\frac{-d}{h} \cdot d}}{\sqrt{-\ell}} \cdot 1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq -1.38 \cdot 10^{-278}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= l -2.5e+143)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l -1.38e-278)
     (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
     (if (<= l 5e-236)
       (/ (* (- d) (sqrt (/ h l))) h)
       (/ d (* (sqrt l) (sqrt 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 (l <= -2.5e+143) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= -1.38e-278) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else if (l <= 5e-236) {
		tmp = (-d * sqrt((h / l))) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: tmp
    if (l <= (-2.5d+143)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (l <= (-1.38d-278)) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else if (l <= 5d-236) then
        tmp = (-d * sqrt((h / l))) / h
    else
        tmp = d / (sqrt(l) * sqrt(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 tmp;
	if (l <= -2.5e+143) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (l <= -1.38e-278) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else if (l <= 5e-236) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	tmp = 0
	if l <= -2.5e+143:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif l <= -1.38e-278:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	elif l <= 5e-236:
		tmp = (-d * math.sqrt((h / l))) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 (l <= -2.5e+143)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= -1.38e-278)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (l <= 5e-236)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (l <= -2.5e+143)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (l <= -1.38e-278)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	elseif (l <= 5e-236)
		tmp = (-d * sqrt((h / l))) / h;
	else
		tmp = d / (sqrt(l) * sqrt(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_] := If[LessEqual[l, -2.5e+143], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.38e-278], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e-236], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.50000000000000006e143

   1. Initial program 51.7%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 61.1

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

   8. Applied rewrites 61.1%

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

   if -2.50000000000000006e143 < l < -1.38000000000000005e-278

   1. Initial program 72.3%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 48.0%

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

   if -1.38000000000000005e-278 < l < 4.9999999999999998e-236

   1. Initial program 81.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in l around -inf

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

   11. Applied rewrites 67.5%

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

   if 4.9999999999999998e-236 < l

   1. Initial program 66.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 42.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 49.4

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq -1.38 \cdot 10^{-278}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{-306}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= l -2.75e-306)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l 5e-236)
     (/ (* (- d) (sqrt (/ h l))) h)
     (/ d (* (sqrt l) (sqrt 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 (l <= -2.75e-306) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 5e-236) {
		tmp = (-d * sqrt((h / l))) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: tmp
    if (l <= (-2.75d-306)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (l <= 5d-236) then
        tmp = (-d * sqrt((h / l))) / h
    else
        tmp = d / (sqrt(l) * sqrt(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 tmp;
	if (l <= -2.75e-306) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (l <= 5e-236) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	tmp = 0
	if l <= -2.75e-306:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif l <= 5e-236:
		tmp = (-d * math.sqrt((h / l))) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 (l <= -2.75e-306)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 5e-236)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (l <= -2.75e-306)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (l <= 5e-236)
		tmp = (-d * sqrt((h / l))) / h;
	else
		tmp = d / (sqrt(l) * sqrt(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_] := If[LessEqual[l, -2.75e-306], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e-236], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.74999999999999996e-306

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 44.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 45.2

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -2.74999999999999996e-306 < l < 4.9999999999999998e-236

   1. Initial program 93.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 13.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Applied rewrites 13.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.7%

       \[\leadsto \frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 4.9999999999999998e-236 < l

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 49.4

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{-306}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 7.2 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

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 (<= h 7.2e-294)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt 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 (h <= 7.2e-294) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: tmp
    if (h <= 7.2d-294) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(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 tmp;
	if (h <= 7.2e-294) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	tmp = 0
	if h <= 7.2e-294:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 (h <= 7.2e-294)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (h <= 7.2e-294)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(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_] := If[LessEqual[h, 7.2e-294], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 7.2000000000000003e-294

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 45.2

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 7.2000000000000003e-294 < h

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 46.6

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 53.7%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 7.2 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 7.5 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (<= h 7.5e-294) (* (- d) (sqrt (pow (* l h) -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 (h <= 7.5e-294) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = 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) :: tmp
    if (h <= 7.5d-294) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = 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 tmp;
	if (h <= 7.5e-294) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = 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):
	tmp = 0
	if h <= 7.5e-294:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = 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)
	tmp = 0.0
	if (h <= 7.5e-294)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = 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)
	tmp = 0.0;
	if (h <= 7.5e-294)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = 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_] := If[LessEqual[h, 7.5e-294], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 7.5000000000000004e-294

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 45.2

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 7.5000000000000004e-294 < h

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 46.6

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 7.5 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.4% 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 67.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 41.5%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. remove-double-neg N/A

      \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 30.5

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 30.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 30.8%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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)))))