Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 77.1%

Time: 15.8s

Alternatives: 18

Speedup: 1.4×

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: 66.5% 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: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{2}{M} \cdot \frac{d}{D\_m}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_0 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_0 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}\right) \cdot {\ell}^{-0.5}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     (*
      (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt (- d)) (sqrt (- l))))
      (fma (/ (* -0.5 (pow (* (/ 2.0 M) (/ d D_m)) -2.0)) l) h 1.0))
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_0 d))
        (pow l -0.5))
       (if (<= d 1.82e-22)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_0 (* d d)))
           d)
          (pow l -0.5))
         (*
          (*
           (*
            (fma (* -0.5 h) (/ (* 0.25 (pow (/ (/ d D_m) M) -2.0)) l) 1.0)
            (sqrt (/ d h)))
           (sqrt d))
          (pow l -0.5)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) / sqrt(-l))) * fma(((-0.5 * pow(((2.0 / M) * (d / D_m)), -2.0)) / l), h, 1.0);
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_0 * d)) * pow(l, -0.5);
	} else if (d <= 1.82e-22) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_0 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = ((fma((-0.5 * h), ((0.25 * pow(((d / D_m) / M), -2.0)) / l), 1.0) * sqrt((d / h))) * sqrt(d)) * pow(l, -0.5);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * fma(Float64(Float64(-0.5 * (Float64(Float64(2.0 / M) * Float64(d / D_m)) ^ -2.0)) / l), h, 1.0));
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_0 * d)) * (l ^ -0.5));
	elseif (d <= 1.82e-22)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_0 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(Float64(d / D_m) / M) ^ -2.0)) / l), 1.0) * sqrt(Float64(d / h))) * sqrt(d)) * (l ^ -0.5));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[Power[N[(N[(2.0 / M), $MachinePrecision] * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.82e-22], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.999999999999994e-310

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 76.4

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

   6. Applied rewrites 76.4%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 1.82e-22

   1. Initial program 63.0%

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]

   if 1.82e-22 < d

   1. Initial program 78.0%

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

   4. Applied rewrites 83.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 93.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}}{\ell}, h, 1\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}\right) \cdot {\ell}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\right) \cdot \left(D\_m \cdot 0.25\right)\right) \cdot \frac{\frac{D\_m}{d}}{d}, M, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l))))
   (if (<= t_0 -5e-116)
     (*
      (*
       (fma (* (* (* (* (/ h l) -0.5) M) (* D_m 0.25)) (/ (/ D_m d) d)) M 1.0)
       (sqrt (/ d h)))
      t_1)
     (if (<= t_0 0.0) (/ d (sqrt (* l h))) (/ t_1 (sqrt (/ h d)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-116) {
		tmp = (fma((((((h / l) * -0.5) * M) * (D_m * 0.25)) * ((D_m / d) / d)), M, 1.0) * sqrt((d / h))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = t_1 / sqrt((h / d));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -5e-116)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(h / l) * -0.5) * M) * Float64(D_m * 0.25)) * Float64(Float64(D_m / d) / d)), M, 1.0) * sqrt(Float64(d / h))) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(t_1 / sqrt(Float64(h / d)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-116], N[(N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M), $MachinePrecision] * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * M + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-116

   1. Initial program 82.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 71.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]

   8. Applied rewrites 65.2%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   10. Applied rewrites 75.1%

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

   if -5.0000000000000003e-116 < (*.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 37.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 72.3%

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

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

   1. Initial program 60.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

   7. Applied rewrites 63.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\right) \cdot \left(D \cdot 0.25\right)\right) \cdot \frac{\frac{D}{d}}{d}, M, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1000000000000:\\ \;\;\;\;\left(\left(\left(\left(\frac{\frac{D\_m \cdot D\_m}{d}}{d} \cdot h\right) \cdot -0.125\right) \cdot \frac{M \cdot M}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l))))
   (if (<= t_0 -1000000000000.0)
     (*
      (*
       (* (* (* (/ (/ (* D_m D_m) d) d) h) -0.125) (/ (* M M) l))
       (sqrt (/ d h)))
      t_1)
     (if (<= t_0 0.0) (/ d (sqrt (* l h))) (/ t_1 (sqrt (/ h d)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double tmp;
	if (t_0 <= -1000000000000.0) {
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * sqrt((d / h))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = t_1 / sqrt((h / d));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / l))
    if (t_0 <= (-1000000000000.0d0)) then
        tmp = (((((((d_m * d_m) / d) / d) * h) * (-0.125d0)) * ((m * m) / l)) * sqrt((d / h))) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = d / sqrt((l * h))
    else
        tmp = t_1 / sqrt((h / d))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -1000000000000.0) {
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * Math.sqrt((d / h))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = t_1 / Math.sqrt((h / d));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -1000000000000.0:
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * math.sqrt((d / h))) * t_1
	elif t_0 <= 0.0:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = t_1 / math.sqrt((h / d))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -1000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) / d) / d) * h) * -0.125) * Float64(Float64(M * M) / l)) * sqrt(Float64(d / h))) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(t_1 / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -1000000000000.0)
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * sqrt((d / h))) * t_1;
	elseif (t_0 <= 0.0)
		tmp = d / sqrt((l * h));
	else
		tmp = t_1 / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1000000000000.0], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e12

   1. Initial program 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 83.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 84.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 69.2%

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

   if -1e12 < (*.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 47.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 56.0%

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

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

   1. Initial program 60.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

   7. Applied rewrites 63.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1000000000000:\\ \;\;\;\;\left(\left(\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125\right) \cdot \frac{M \cdot M}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 81.7%

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

   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. Initial program 0.0%

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

   4. Applied rewrites 2.2%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

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

Alternative 5: 72.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 81.7%

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

   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. Initial program 0.0%

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

   4. Applied rewrites 2.2%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

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

Alternative 6: 71.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 81.7%

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

   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. Initial program 0.0%

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

   4. Applied rewrites 2.2%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 24.3

          \[\leadsto \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{\sqrt{h}}{d} \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right) \cdot {\ell}^{-0.5} \]

   7. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{\sqrt{h}}{d} \cdot \frac{M \cdot M}{\ell}\right)\right)} \cdot {\ell}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

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

Alternative 7: 69.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 81.7%

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

   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. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 2.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 20.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125, \frac{M \cdot M}{\ell}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

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

Alternative 8: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D\_m}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_0 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_0 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}\right) \cdot {\ell}^{-0.5}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     (/
      (*
       (*
        (fma (* -0.5 (/ h l)) (pow (* (/ 2.0 M) (/ d D_m)) -2.0) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_0 d))
        (pow l -0.5))
       (if (<= d 1.82e-22)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_0 (* d d)))
           d)
          (pow l -0.5))
         (*
          (*
           (*
            (fma (* -0.5 h) (/ (* 0.25 (pow (/ (/ d D_m) M) -2.0)) l) 1.0)
            (sqrt (/ d h)))
           (sqrt d))
          (pow l -0.5)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((2.0 / M) * (d / D_m)), -2.0), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_0 * d)) * pow(l, -0.5);
	} else if (d <= 1.82e-22) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_0 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = ((fma((-0.5 * h), ((0.25 * pow(((d / D_m) / M), -2.0)) / l), 1.0) * sqrt((d / h))) * sqrt(d)) * pow(l, -0.5);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(2.0 / M) * Float64(d / D_m)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_0 * d)) * (l ^ -0.5));
	elseif (d <= 1.82e-22)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_0 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(Float64(d / D_m) / M) ^ -2.0)) / l), 1.0) * sqrt(Float64(d / h))) * sqrt(d)) * (l ^ -0.5));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 / M), $MachinePrecision] * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.82e-22], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.999999999999994e-310

   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

   4. Applied rewrites 71.2%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 1.82e-22

   1. Initial program 63.0%

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]

   if 1.82e-22 < d

   1. Initial program 78.0%

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

   4. Applied rewrites 83.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 93.1%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}\right) \cdot {\ell}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D\_m}\right)}^{-2}, 1\right) \cdot t\_0\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_1 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_1 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     (/
      (*
       (* (fma (* -0.5 (/ h l)) (pow (* (/ 2.0 M) (/ d D_m)) -2.0) 1.0) t_0)
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_1 d))
        (pow l -0.5))
       (if (<= d 4.4e-36)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_1 (* d d)))
           d)
          (pow l -0.5))
         (*
          (*
           (fma (* -0.5 h) (/ (* 0.25 (pow (/ (/ d D_m) M) -2.0)) l) 1.0)
           (sqrt (/ d h)))
          t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((2.0 / M) * (d / D_m)), -2.0), 1.0) * t_0) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_1 * d)) * pow(l, -0.5);
	} else if (d <= 4.4e-36) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_1 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = (fma((-0.5 * h), ((0.25 * pow(((d / D_m) / M), -2.0)) / l), 1.0) * sqrt((d / h))) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(2.0 / M) * Float64(d / D_m)) ^ -2.0), 1.0) * t_0) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_1 * d)) * (l ^ -0.5));
	elseif (d <= 4.4e-36)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_1 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(Float64(d / D_m) / M) ^ -2.0)) / l), 1.0) * sqrt(Float64(d / h))) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 / M), $MachinePrecision] * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-36], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.999999999999994e-310

   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

   4. Applied rewrites 71.2%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 4.3999999999999999e-36

   1. Initial program 58.8%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]

   if 4.3999999999999999e-36 < d

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 88.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D\_m}\right)}^{-2}, 1\right) \cdot t\_0\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_1 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_1 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     (/
      (*
       (* (fma (* -0.5 (/ h l)) (pow (* (/ 2.0 M) (/ d D_m)) -2.0) 1.0) t_0)
       (sqrt (- d)))
      (sqrt (- l)))
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_1 d))
        (pow l -0.5))
       (if (<= d 4.4e-36)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_1 (* d d)))
           d)
          (pow l -0.5))
         (*
          (*
           (fma (* -0.5 h) (/ (* 0.25 (pow (/ (/ d D_m) M) -2.0)) l) 1.0)
           t_0)
          (sqrt (/ d l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((2.0 / M) * (d / D_m)), -2.0), 1.0) * t_0) * sqrt(-d)) / sqrt(-l);
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_1 * d)) * pow(l, -0.5);
	} else if (d <= 4.4e-36) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_1 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = (fma((-0.5 * h), ((0.25 * pow(((d / D_m) / M), -2.0)) / l), 1.0) * t_0) * sqrt((d / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(2.0 / M) * Float64(d / D_m)) ^ -2.0), 1.0) * t_0) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_1 * d)) * (l ^ -0.5));
	elseif (d <= 4.4e-36)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_1 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(Float64(d / D_m) / M) ^ -2.0)) / l), 1.0) * t_0) * sqrt(Float64(d / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 / M), $MachinePrecision] * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-36], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.999999999999994e-310

   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

   4. Applied rewrites 70.9%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 4.3999999999999999e-36

   1. Initial program 58.8%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]

   if 4.3999999999999999e-36 < d

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 88.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{d}{M \cdot D\_m}\right)}^{-2}}{\ell}, 1\right) \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_2 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_2 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))) (t_2 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     (*
      (* (fma (* -0.5 h) (/ (* 0.25 (pow (/ d (* M D_m)) -2.0)) l) 1.0) t_0)
      t_1)
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_2 d))
        (pow l -0.5))
       (if (<= d 4.4e-36)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_2 (* d d)))
           d)
          (pow l -0.5))
         (*
          (*
           (fma (* -0.5 h) (/ (* 0.25 (pow (/ (/ d D_m) M) -2.0)) l) 1.0)
           t_0)
          t_1))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = (fma((-0.5 * h), ((0.25 * pow((d / (M * D_m)), -2.0)) / l), 1.0) * t_0) * t_1;
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_2 * d)) * pow(l, -0.5);
	} else if (d <= 4.4e-36) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_2 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = (fma((-0.5 * h), ((0.25 * pow(((d / D_m) / M), -2.0)) / l), 1.0) * t_0) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(d / Float64(M * D_m)) ^ -2.0)) / l), 1.0) * t_0) * t_1);
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_2 * d)) * (l ^ -0.5));
	elseif (d <= 4.4e-36)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_2 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(Float64(d / D_m) / M) ^ -2.0)) / l), 1.0) * t_0) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(d / N[(M * D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-36], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.999999999999994e-310

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 65.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 69.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 4.3999999999999999e-36

   1. Initial program 58.8%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]

   if 4.3999999999999999e-36 < d

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 88.4%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{d}{M \cdot D\_m}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{{h}^{-1}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D\_m \cdot D\_m}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, t\_1 \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell}, t\_1 \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (fma (* -0.5 h) (/ (* 0.25 (pow (/ d (* M D_m)) -2.0)) l) 1.0)
           (sqrt (/ d h)))
          (sqrt (/ d l))))
        (t_1 (sqrt (pow h -1.0))))
   (if (<= d -2e-310)
     t_0
     (if (<= d 2.7e-151)
       (*
        (fma (* (* M M) (* (/ (/ (* D_m D_m) l) d) -0.125)) (sqrt h) (* t_1 d))
        (pow l -0.5))
       (if (<= d 4.4e-36)
         (*
          (/
           (fma
            (* -0.125 (sqrt h))
            (* (* (* M M) D_m) (/ D_m l))
            (* t_1 (* d d)))
           d)
          (pow l -0.5))
         t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (fma((-0.5 * h), ((0.25 * pow((d / (M * D_m)), -2.0)) / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	double t_1 = sqrt(pow(h, -1.0));
	double tmp;
	if (d <= -2e-310) {
		tmp = t_0;
	} else if (d <= 2.7e-151) {
		tmp = fma(((M * M) * ((((D_m * D_m) / l) / d) * -0.125)), sqrt(h), (t_1 * d)) * pow(l, -0.5);
	} else if (d <= 4.4e-36) {
		tmp = (fma((-0.125 * sqrt(h)), (((M * M) * D_m) * (D_m / l)), (t_1 * (d * d))) / d) * pow(l, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(fma(Float64(-0.5 * h), Float64(Float64(0.25 * (Float64(d / Float64(M * D_m)) ^ -2.0)) / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)))
	t_1 = sqrt((h ^ -1.0))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = t_0;
	elseif (d <= 2.7e-151)
		tmp = Float64(fma(Float64(Float64(M * M) * Float64(Float64(Float64(Float64(D_m * D_m) / l) / d) * -0.125)), sqrt(h), Float64(t_1 * d)) * (l ^ -0.5));
	elseif (d <= 4.4e-36)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(Float64(Float64(M * M) * D_m) * Float64(D_m / l)), Float64(t_1 * Float64(d * d))) / d) * (l ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(0.25 * N[Power[N[(d / N[(M * D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-310], t$95$0, If[LessEqual[d, 2.7e-151], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-36], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310 or 4.3999999999999999e-36 < d

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 76.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.4

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

   8. Applied rewrites 76.4%

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

   if -1.999999999999994e-310 < d < 2.70000000000000007e-151

   1. Initial program 42.3%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in l around inf

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

   7. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)} \cdot {\ell}^{-0.5} \]

   if 2.70000000000000007e-151 < d < 4.3999999999999999e-36

   1. Initial program 58.8%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{\frac{1}{h}} \cdot \left(d \cdot d\right)\right)}{d}} \cdot {\ell}^{-0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{\frac{D \cdot D}{\ell}}{d} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell}, \sqrt{{h}^{-1}} \cdot \left(d \cdot d\right)\right)}{d} \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot h, \frac{0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M \cdot D\_m \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;M \cdot D\_m \leq 6 \cdot 10^{+205}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D\_m \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D\_m \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\frac{D\_m \cdot D\_m}{d}}{d} \cdot h\right) \cdot -0.125\right) \cdot \frac{M \cdot M}{\ell}\right) \cdot t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))))
   (if (<= (* M D_m) 5e-94)
     (/ t_1 (sqrt (/ h d)))
     (if (<= (* M D_m) 6e+205)
       (*
        (* t_0 t_1)
        (-
         1.0
         (/ (* (* (/ h l) 0.5) (* D_m M)) (/ (* 4.0 (* d d)) (* D_m M)))))
       (*
        (* (* (* (* (/ (/ (* D_m D_m) d) d) h) -0.125) (/ (* M M) l)) t_0)
        t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double tmp;
	if ((M * D_m) <= 5e-94) {
		tmp = t_1 / sqrt((h / d));
	} else if ((M * D_m) <= 6e+205) {
		tmp = (t_0 * t_1) * (1.0 - ((((h / l) * 0.5) * (D_m * M)) / ((4.0 * (d * d)) / (D_m * M))));
	} else {
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * t_0) * t_1;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    if ((m * d_m) <= 5d-94) then
        tmp = t_1 / sqrt((h / d))
    else if ((m * d_m) <= 6d+205) then
        tmp = (t_0 * t_1) * (1.0d0 - ((((h / l) * 0.5d0) * (d_m * m)) / ((4.0d0 * (d * d)) / (d_m * m))))
    else
        tmp = (((((((d_m * d_m) / d) / d) * h) * (-0.125d0)) * ((m * m) / l)) * t_0) * t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if ((M * D_m) <= 5e-94) {
		tmp = t_1 / Math.sqrt((h / d));
	} else if ((M * D_m) <= 6e+205) {
		tmp = (t_0 * t_1) * (1.0 - ((((h / l) * 0.5) * (D_m * M)) / ((4.0 * (d * d)) / (D_m * M))));
	} else {
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * t_0) * t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if (M * D_m) <= 5e-94:
		tmp = t_1 / math.sqrt((h / d))
	elif (M * D_m) <= 6e+205:
		tmp = (t_0 * t_1) * (1.0 - ((((h / l) * 0.5) * (D_m * M)) / ((4.0 * (d * d)) / (D_m * M))))
	else:
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * t_0) * t_1
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (Float64(M * D_m) <= 5e-94)
		tmp = Float64(t_1 / sqrt(Float64(h / d)));
	elseif (Float64(M * D_m) <= 6e+205)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(D_m * M)) / Float64(Float64(4.0 * Float64(d * d)) / Float64(D_m * M)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) / d) / d) * h) * -0.125) * Float64(Float64(M * M) / l)) * t_0) * t_1);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if ((M * D_m) <= 5e-94)
		tmp = t_1 / sqrt((h / d));
	elseif ((M * D_m) <= 6e+205)
		tmp = (t_0 * t_1) * (1.0 - ((((h / l) * 0.5) * (D_m * M)) / ((4.0 * (d * d)) / (D_m * M))));
	else
		tmp = (((((((D_m * D_m) / d) / d) * h) * -0.125) * ((M * M) / l)) * t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M * D$95$m), $MachinePrecision], 5e-94], N[(t$95$1 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * D$95$m), $MachinePrecision], 6e+205], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(D$95$m * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.9999999999999995e-94

   1. Initial program 65.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.3

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

   5. Applied rewrites 36.3%

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

   7. Applied rewrites 46.5%

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

   if 4.9999999999999995e-94 < (*.f64 M D) < 5.9999999999999999e205

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 71.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if 5.9999999999999999e205 < (*.f64 M D)

   1. Initial program 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 64.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 78.3%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 77.9%

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

Alternative 14: 57.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D\_m}{\frac{d \cdot d}{M}} \cdot M}{4} \cdot D\_m, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -5e-310)
   (/ (/ (- d) (sqrt (- l))) (sqrt (- h)))
   (if (<= l 1.2e+129)
     (*
      (fma (* (/ (* (/ D_m (/ (* d d) M)) M) 4.0) D_m) (* (/ h l) -0.5) 1.0)
      (/ d (sqrt (* l h))))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d / sqrt(-l)) / sqrt(-h);
	} else if (l <= 1.2e+129) {
		tmp = fma(((((D_m / ((d * d) / M)) * M) / 4.0) * D_m), ((h / l) * -0.5), 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-l))) / sqrt(Float64(-h)));
	elseif (l <= 1.2e+129)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m / Float64(Float64(d * d) / M)) * M) / 4.0) * D_m), Float64(Float64(h / l) * -0.5), 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[((-d) / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+129], N[(N[(N[(N[(N[(N[(D$95$m / N[(N[(d * d), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] / 4.0), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 9.8

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

   5. Applied rewrites 9.8%

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

   7. Applied rewrites 33.1%

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

   9. Applied rewrites 45.5%

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

   if -4.999999999999985e-310 < l < 1.1999999999999999e129

   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. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 63.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 63.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]

   6. Applied rewrites 63.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot M}}\right) \]

   8. Applied rewrites 59.0%

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

   10. Applied rewrites 64.7%

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

   if 1.1999999999999999e129 < l

   1. Initial program 55.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 58.2%

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

   9. Applied rewrites 67.9%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D}{\frac{d \cdot d}{M}} \cdot M}{4} \cdot D, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -5e-310)
   (/ (/ (- d) (sqrt (- l))) (sqrt (- h)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d / sqrt(-l)) / sqrt(-h);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (-d / sqrt(-l)) / sqrt(-h)
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d / Math.sqrt(-l)) / Math.sqrt(-h);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -5e-310:
		tmp = (-d / math.sqrt(-l)) / math.sqrt(-h)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-l))) / sqrt(Float64(-h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (-d / sqrt(-l)) / sqrt(-h);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[((-d) / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 9.8

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

   5. Applied rewrites 9.8%

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

   7. Applied rewrites 33.1%

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

   9. Applied rewrites 45.5%

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

   if -4.999999999999985e-310 < l

   1. Initial program 67.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   9. Applied rewrites 57.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l 7.8e-259) (/ (- d) (sqrt (* l h))) (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.8e-259) {
		tmp = -d / sqrt((l * h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 7.8d-259) then
        tmp = -d / sqrt((l * h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.8e-259) {
		tmp = -d / Math.sqrt((l * h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= 7.8e-259:
		tmp = -d / math.sqrt((l * h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= 7.8e-259)
		tmp = Float64(Float64(-d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= 7.8e-259)
		tmp = -d / sqrt((l * h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, 7.8e-259], N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.80000000000000031e-259

   1. Initial program 66.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   5. Applied rewrites 11.3%

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

   7. Applied rewrites 11.3%

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

   9. Applied rewrites 39.9%

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

   if 7.80000000000000031e-259 < l

   1. Initial program 66.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   7. Applied rewrites 53.7%

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

   9. Applied rewrites 59.3%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= d -3.4e-171) (/ (- d) t_0) (/ d t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -3.4e-171) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= (-3.4d-171)) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= -3.4e-171) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= -3.4e-171:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -3.4e-171)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -3.4e-171)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.4e-171], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.39999999999999985e-171

   1. Initial program 73.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 7.4

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

   5. Applied rewrites 7.4%

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

   7. Applied rewrites 7.4%

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

   9. Applied rewrites 46.9%

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

   if -3.39999999999999985e-171 < d

   1. Initial program 61.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   7. Applied rewrites 46.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

3. Taylor expanded in d around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 30.8

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

5. Applied rewrites 30.8%

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

7. Applied rewrites 30.9%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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)))))