Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 79.2%

Time: 15.8s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{{\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}}{\ell}, \frac{-0.5}{{h}^{-1}}, 1\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{\frac{\frac{M\_m \cdot M\_m}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -9.2e-303)
   (*
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
    (fma (/ (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0) l) (/ -0.5 (pow h -1.0)) 1.0))
   (if (<= d 4.4e-216)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
      (sqrt l))
     (/
      (*
       (+
        (sqrt (pow h -1.0))
        (* (* -0.125 (sqrt h)) (* D_m (* D_m (/ (/ (/ (* M_m M_m) d) d) l)))))
       d)
      (sqrt l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.2e-303) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * fma((pow(((d / M_m) * (2.0 / D_m)), -2.0) / l), (-0.5 / pow(h, -1.0)), 1.0);
	} else if (d <= 4.4e-216) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = ((sqrt(pow(h, -1.0)) + ((-0.125 * sqrt(h)) * (D_m * (D_m * ((((M_m * M_m) / d) / d) / l))))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -9.2e-303)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * fma(Float64((Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0) / l), Float64(-0.5 / (h ^ -1.0)), 1.0));
	elseif (d <= 4.4e-216)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt((h ^ -1.0)) + Float64(Float64(-0.125 * sqrt(h)) * Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) / l))))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -9.2e-303], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision] * N[(-0.5 / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-216], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.19999999999999981e-303

   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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if -9.19999999999999981e-303 < d < 4.3999999999999998e-216

   1. Initial program 18.0%

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

   4. Applied rewrites 26.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 74.0%

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

   if 4.3999999999999998e-216 < d

   1. Initial program 71.6%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 78.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{{\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}{\ell}, \frac{-0.5}{{h}^{-1}}, 1\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D \cdot \left(D \cdot \frac{\frac{\frac{M \cdot M}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.4%

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 84.7

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

   6. Applied rewrites 84.7%

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

   8. Applied rewrites 84.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 84.7%

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

   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 1.8%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 25.9

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

   7. Applied rewrites 25.9%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 36.6%

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

4. Final simplification 74.4%

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

Alternative 3: 50.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-172)
   (* (* (* 0.125 (* D_m D_m)) (/ (* M_m M_m) d)) (/ (sqrt (/ h l)) (fabs l)))
   (/ (sqrt (/ d h)) (sqrt (/ l d)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double 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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-172) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / fabs(l));
	} else {
		tmp = sqrt((d / h)) / sqrt((l / d));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-1d-172)) then
        tmp = ((0.125d0 * (d_m * d_m)) * ((m_m * m_m) / d)) * (sqrt((h / l)) / abs(l))
    else
        tmp = sqrt((d / h)) / sqrt((l / d))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-172) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else {
		tmp = Math.sqrt((d / h)) / Math.sqrt((l / d));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-172:
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (math.sqrt((h / l)) / math.fabs(l))
	else:
		tmp = math.sqrt((d / h)) / math.sqrt((l / d))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, 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_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-172)
		tmp = Float64(Float64(Float64(0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * M_m) / d)) * Float64(sqrt(Float64(h / l)) / abs(l)));
	else
		tmp = Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-172)
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / abs(l));
	else
		tmp = sqrt((d / h)) / sqrt((l / d));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-172], N[(N[(N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-172

   1. Initial program 89.2%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 39.6%

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

   7. Applied rewrites 44.3%

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

   if -1e-172 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.3%

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

   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 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 63.0%

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

4. Final simplification 58.2%

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

Alternative 4: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M\_m} \cdot \frac{2}{D\_m}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{\frac{\frac{M\_m \cdot M\_m}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (/
    (*
     (*
      (fma (* -0.5 (/ h l)) (pow (* (/ d M_m) (/ 2.0 D_m)) -2.0) 1.0)
      (sqrt (/ d l)))
     (sqrt (- d)))
    (sqrt (- h)))
   (if (<= d 4.4e-216)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
      (sqrt l))
     (/
      (*
       (+
        (sqrt (pow h -1.0))
        (* (* -0.125 (sqrt h)) (* D_m (* D_m (/ (/ (/ (* M_m M_m) d) d) l)))))
       d)
      (sqrt l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M_m) * (2.0 / D_m)), -2.0), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 4.4e-216) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = ((sqrt(pow(h, -1.0)) + ((-0.125 * sqrt(h)) * (D_m * (D_m * ((((M_m * M_m) / d) / d) / l))))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M_m) * Float64(2.0 / D_m)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 4.4e-216)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt((h ^ -1.0)) + Float64(Float64(-0.125 * sqrt(h)) * Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) / l))))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / 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, 4.4e-216], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 68.7%

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

   4. Applied rewrites 77.0%

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

   if -4.999999999999985e-310 < d < 4.3999999999999998e-216

   1. Initial program 20.7%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 85.1%

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

   if 4.3999999999999998e-216 < d

   1. Initial program 71.6%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 78.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D \cdot \left(D \cdot \frac{\frac{\frac{M \cdot M}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D\_m \cdot M\_m}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{\frac{\frac{M\_m \cdot M\_m}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (*
     (fma (* (/ h l) -0.5) (* 0.25 (pow (/ d (* D_m M_m)) -2.0)) 1.0)
     (sqrt (/ d l)))
    (/ (sqrt (- d)) (sqrt (- h))))
   (if (<= d 4.4e-216)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
      (sqrt l))
     (/
      (*
       (+
        (sqrt (pow h -1.0))
        (* (* -0.125 (sqrt h)) (* D_m (* D_m (/ (/ (/ (* M_m M_m) d) d) l)))))
       d)
      (sqrt l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (fma(((h / l) * -0.5), (0.25 * pow((d / (D_m * M_m)), -2.0)), 1.0) * sqrt((d / l))) * (sqrt(-d) / sqrt(-h));
	} else if (d <= 4.4e-216) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = ((sqrt(pow(h, -1.0)) + ((-0.125 * sqrt(h)) * (D_m * (D_m * ((((M_m * M_m) / d) / d) / l))))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), Float64(0.25 * (Float64(d / Float64(D_m * M_m)) ^ -2.0)), 1.0) * sqrt(Float64(d / l))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))));
	elseif (d <= 4.4e-216)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt((h ^ -1.0)) + Float64(Float64(-0.125 * sqrt(h)) * Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) / l))))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.25 * N[Power[N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-216], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 68.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 72.0

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

   6. Applied rewrites 72.0%

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

   8. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 77.0

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

   10. Applied rewrites 77.0%

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

   if -4.999999999999985e-310 < d < 4.3999999999999998e-216

   1. Initial program 20.7%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 85.1%

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

   if 4.3999999999999998e-216 < d

   1. Initial program 71.6%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 78.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, 0.25 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D \cdot \left(D \cdot \frac{\frac{\frac{M \cdot M}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.125 \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m}\right)}^{-2}}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{\frac{\frac{M\_m \cdot M\_m}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -3.7e-296)
   (*
    (*
     (fma (/ (* -0.125 (pow (/ (/ d M_m) D_m) -2.0)) l) h 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (if (<= d 4.4e-216)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
      (sqrt l))
     (/
      (*
       (+
        (sqrt (pow h -1.0))
        (* (* -0.125 (sqrt h)) (* D_m (* D_m (/ (/ (/ (* M_m M_m) d) d) l)))))
       d)
      (sqrt l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -3.7e-296) {
		tmp = (fma(((-0.125 * pow(((d / M_m) / D_m), -2.0)) / l), h, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= 4.4e-216) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = ((sqrt(pow(h, -1.0)) + ((-0.125 * sqrt(h)) * (D_m * (D_m * ((((M_m * M_m) / d) / d) / l))))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -3.7e-296)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * (Float64(Float64(d / M_m) / D_m) ^ -2.0)) / l), h, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= 4.4e-216)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt((h ^ -1.0)) + Float64(Float64(-0.125 * sqrt(h)) * Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) / l))))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -3.7e-296], N[(N[(N[(N[(N[(-0.125 * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / D$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-216], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.70000000000000027e-296

   1. Initial program 71.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 74.4

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

   6. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 74.4

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

   8. Applied rewrites 74.4%

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

   10. Applied rewrites 74.4%

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

   if -3.70000000000000027e-296 < d < 4.3999999999999998e-216

   1. Initial program 17.3%

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

   4. Applied rewrites 25.3%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 50.7

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 71.0%

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

   if 4.3999999999999998e-216 < d

   1. Initial program 71.6%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 78.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.125 \cdot {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D \cdot \left(D \cdot \frac{\frac{\frac{M \cdot M}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot t\_0, t\_0, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{\frac{\frac{M\_m \cdot M\_m}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= d -9.2e-303)
     (*
      (* (fma (* (* (/ h l) -0.125) t_0) t_0 1.0) (sqrt (/ d l)))
      (sqrt (/ d h)))
     (if (<= d 4.4e-216)
       (/
        (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
        (sqrt l))
       (/
        (*
         (+
          (sqrt (pow h -1.0))
          (*
           (* -0.125 (sqrt h))
           (* D_m (* D_m (/ (/ (/ (* M_m M_m) d) d) l)))))
         d)
        (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (d <= -9.2e-303) {
		tmp = (fma((((h / l) * -0.125) * t_0), t_0, 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else if (d <= 4.4e-216) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = ((sqrt(pow(h, -1.0)) + ((-0.125 * sqrt(h)) * (D_m * (D_m * ((((M_m * M_m) / d) / d) / l))))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= -9.2e-303)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.125) * t_0), t_0, 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (d <= 4.4e-216)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt((h ^ -1.0)) + Float64(Float64(-0.125 * sqrt(h)) * Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) / l))))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.2e-303], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-216], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.19999999999999981e-303

   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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 73.8

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

   6. Applied rewrites 73.8%

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

   8. Applied rewrites 70.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 71.7%

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

   if -9.19999999999999981e-303 < d < 4.3999999999999998e-216

   1. Initial program 18.0%

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

   4. Applied rewrites 26.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 74.0%

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

   if 4.3999999999999998e-216 < d

   1. Initial program 71.6%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 78.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.125\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot \frac{M}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{h}^{-1}} + \left(-0.125 \cdot \sqrt{h}\right) \cdot \left(D \cdot \left(D \cdot \frac{\frac{\frac{M \cdot M}{d}}{d}}{\ell}\right)\right)\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot M\_m}{d}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot t\_0\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot \left(\frac{t\_0}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m M_m) d)))
   (if (<= d -2.9e-52)
     (/ (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ l d)))
     (if (<= d -8e-309)
       (* (* (* 0.125 (* D_m D_m)) t_0) (/ (sqrt (/ h l)) (fabs l)))
       (if (<= d 9e-145)
         (/
          (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
          (sqrt l))
         (if (<= d 3.5e+151)
           (/
            (fma
             (* (* D_m D_m) (* (/ t_0 l) -0.125))
             (sqrt h)
             (* (sqrt (pow h -1.0)) d))
            (sqrt l))
           (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * M_m) / d;
	double tmp;
	if (d <= -2.9e-52) {
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	} else if (d <= -8e-309) {
		tmp = ((0.125 * (D_m * D_m)) * t_0) * (sqrt((h / l)) / fabs(l));
	} else if (d <= 9e-145) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else if (d <= 3.5e+151) {
		tmp = fma(((D_m * D_m) * ((t_0 / l) * -0.125)), sqrt(h), (sqrt(pow(h, -1.0)) * d)) / sqrt(l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * M_m) / d)
	tmp = 0.0
	if (d <= -2.9e-52)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) / sqrt(Float64(l / d)));
	elseif (d <= -8e-309)
		tmp = Float64(Float64(Float64(0.125 * Float64(D_m * D_m)) * t_0) * Float64(sqrt(Float64(h / l)) / abs(l)));
	elseif (d <= 9e-145)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	elseif (d <= 3.5e+151)
		tmp = Float64(fma(Float64(Float64(D_m * D_m) * Float64(Float64(t_0 / l) * -0.125)), sqrt(h), Float64(sqrt((h ^ -1.0)) * d)) / sqrt(l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.9e-52], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-309], N[(N[(N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e-145], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+151], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(t$95$0 / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.9000000000000002e-52

   1. Initial program 80.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 68.0%

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

   if -2.9000000000000002e-52 < d < -8.0000000000000003e-309

   1. Initial program 53.5%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 40.9%

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

   if -8.0000000000000003e-309 < d < 9.0000000000000001e-145

   1. Initial program 25.6%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 78.2%

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

   if 9.0000000000000001e-145 < d < 3.5000000000000003e151

   1. Initial program 72.6%

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 73.4%

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

   if 3.5000000000000003e151 < 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. 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 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 90.6%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot \sqrt{\ell \cdot h}\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d 6.5e-176)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ (* M_m M_m) d) (sqrt (* l h))))
      (* l l))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= 6.5e-176) {
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * sqrt((l * h)))) / (l * l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5d-310)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (d <= 6.5d-176) then
        tmp = (((d_m * d_m) * (-0.125d0)) * (((m_m * m_m) / d) * sqrt((l * h)))) / (l * l)
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (d <= 6.5e-176) {
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * Math.sqrt((l * h)))) / (l * l);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5e-310:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif d <= 6.5e-176:
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * math.sqrt((l * h)))) / (l * l)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= 6.5e-176)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(Float64(M_m * M_m) / d) * sqrt(Float64(l * h)))) / Float64(l * l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (d <= 6.5e-176)
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * sqrt((l * h)))) / (l * l);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e-176], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 68.7%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

   if -4.999999999999985e-310 < d < 6.5e-176

   1. Initial program 23.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in d around 0

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

   8. Applied rewrites 43.0%

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

   if 6.5e-176 < d

   1. Initial program 73.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 68.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M \cdot M}{d} \cdot \sqrt{\ell \cdot h}\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 4.5e-173)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.5e-173) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.5d-173) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.5e-173) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 4.5e-173:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 4.5e-173)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 4.5e-173)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 4.5e-173], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.50000000000000018e-173

   1. Initial program 60.4%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 4.50000000000000018e-173 < d

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 59.1%

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

   9. Applied rewrites 68.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{-210}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= d -6.6e-210) (* (- d) t_0) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (d <= -6.6e-210) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (d <= (-6.6d-210)) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (d <= -6.6e-210) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if d <= -6.6e-210:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (d <= -6.6e-210)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (d <= -6.6e-210)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.6e-210], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.6e-210

   1. Initial program 75.9%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -6.6e-210 < d

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.2

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

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-210}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* (sqrt (pow (* l h) -1.0)) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt(pow((l * h), -1.0)) * d;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.sqrt(math.pow((l * h), -1.0)) * d

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(sqrt((Float64(l * h) ^ -1.0)) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = sqrt(((l * h) ^ -1.0)) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 31.3

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

5. Applied rewrites 31.3%

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

6. Final simplification 31.3%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
7. Add Preprocessing

Alternative 13: 58.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m}{\ell} \cdot \frac{M\_m \cdot \sqrt{h}}{d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.9e-52)
   (/ (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ l d)))
   (if (<= d -8e-309)
     (*
      (* (* 0.125 (* D_m D_m)) (/ (* M_m M_m) d))
      (/ (sqrt (/ h l)) (fabs l)))
     (if (<= d 1.3e-144)
       (/
        (* (* (* D_m D_m) -0.125) (* (/ M_m l) (/ (* M_m (sqrt h)) d)))
        (sqrt l))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-52) {
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	} else if (d <= -8e-309) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / fabs(l));
	} else if (d <= 1.3e-144) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.9d-52)) then
        tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d))
    else if (d <= (-8d-309)) then
        tmp = ((0.125d0 * (d_m * d_m)) * ((m_m * m_m) / d)) * (sqrt((h / l)) / abs(l))
    else if (d <= 1.3d-144) then
        tmp = (((d_m * d_m) * (-0.125d0)) * ((m_m / l) * ((m_m * sqrt(h)) / d))) / sqrt(l)
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-52) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) / Math.sqrt((l / d));
	} else if (d <= -8e-309) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else if (d <= 1.3e-144) {
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * Math.sqrt(h)) / d))) / Math.sqrt(l);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.9e-52:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) / math.sqrt((l / d))
	elif d <= -8e-309:
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (math.sqrt((h / l)) / math.fabs(l))
	elif d <= 1.3e-144:
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * math.sqrt(h)) / d))) / math.sqrt(l)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.9e-52)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) / sqrt(Float64(l / d)));
	elseif (d <= -8e-309)
		tmp = Float64(Float64(Float64(0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * M_m) / d)) * Float64(sqrt(Float64(h / l)) / abs(l)));
	elseif (d <= 1.3e-144)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(M_m / l) * Float64(Float64(M_m * sqrt(h)) / d))) / sqrt(l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.9e-52)
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	elseif (d <= -8e-309)
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / abs(l));
	elseif (d <= 1.3e-144)
		tmp = (((D_m * D_m) * -0.125) * ((M_m / l) * ((M_m * sqrt(h)) / d))) / sqrt(l);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.9e-52], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-309], N[(N[(N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-144], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.9000000000000002e-52

   1. Initial program 80.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 68.0%

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

   if -2.9000000000000002e-52 < d < -8.0000000000000003e-309

   1. Initial program 53.5%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 40.9%

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

   if -8.0000000000000003e-309 < d < 1.3e-144

   1. Initial program 25.6%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 78.2%

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

   if 1.3e-144 < d

   1. Initial program 75.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 70.4%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(M\_m \cdot \frac{\sqrt{h} \cdot M\_m}{\ell \cdot d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.9e-52)
   (/ (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ l d)))
   (if (<= d -8e-309)
     (*
      (* (* 0.125 (* D_m D_m)) (/ (* M_m M_m) d))
      (/ (sqrt (/ h l)) (fabs l)))
     (if (<= d 1.3e-144)
       (/
        (* (* (* D_m D_m) -0.125) (* M_m (/ (* (sqrt h) M_m) (* l d))))
        (sqrt l))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-52) {
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	} else if (d <= -8e-309) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / fabs(l));
	} else if (d <= 1.3e-144) {
		tmp = (((D_m * D_m) * -0.125) * (M_m * ((sqrt(h) * M_m) / (l * d)))) / sqrt(l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.9d-52)) then
        tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d))
    else if (d <= (-8d-309)) then
        tmp = ((0.125d0 * (d_m * d_m)) * ((m_m * m_m) / d)) * (sqrt((h / l)) / abs(l))
    else if (d <= 1.3d-144) then
        tmp = (((d_m * d_m) * (-0.125d0)) * (m_m * ((sqrt(h) * m_m) / (l * d)))) / sqrt(l)
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-52) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) / Math.sqrt((l / d));
	} else if (d <= -8e-309) {
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else if (d <= 1.3e-144) {
		tmp = (((D_m * D_m) * -0.125) * (M_m * ((Math.sqrt(h) * M_m) / (l * d)))) / Math.sqrt(l);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.9e-52:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) / math.sqrt((l / d))
	elif d <= -8e-309:
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (math.sqrt((h / l)) / math.fabs(l))
	elif d <= 1.3e-144:
		tmp = (((D_m * D_m) * -0.125) * (M_m * ((math.sqrt(h) * M_m) / (l * d)))) / math.sqrt(l)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.9e-52)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) / sqrt(Float64(l / d)));
	elseif (d <= -8e-309)
		tmp = Float64(Float64(Float64(0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * M_m) / d)) * Float64(sqrt(Float64(h / l)) / abs(l)));
	elseif (d <= 1.3e-144)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(M_m * Float64(Float64(sqrt(h) * M_m) / Float64(l * d)))) / sqrt(l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.9e-52)
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	elseif (d <= -8e-309)
		tmp = ((0.125 * (D_m * D_m)) * ((M_m * M_m) / d)) * (sqrt((h / l)) / abs(l));
	elseif (d <= 1.3e-144)
		tmp = (((D_m * D_m) * -0.125) * (M_m * ((sqrt(h) * M_m) / (l * d)))) / sqrt(l);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.9e-52], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-309], N[(N[(N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-144], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(N[Sqrt[h], $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.9000000000000002e-52

   1. Initial program 80.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 68.0%

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

   if -2.9000000000000002e-52 < d < -8.0000000000000003e-309

   1. Initial program 53.5%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 40.9%

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

   if -8.0000000000000003e-309 < d < 1.3e-144

   1. Initial program 25.6%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

   9. Applied rewrites 66.0%

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

   11. Applied rewrites 78.2%

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

   if 1.3e-144 < d

   1. Initial program 75.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 70.4%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot \sqrt{\ell \cdot h}\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (/ (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ l d)))
   (if (<= d 6.5e-176)
     (/
      (* (* (* D_m D_m) -0.125) (* (/ (* M_m M_m) d) (sqrt (* l h))))
      (* l l))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	} else if (d <= 6.5e-176) {
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * sqrt((l * h)))) / (l * l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d))
    else if (d <= 6.5d-176) then
        tmp = (((d_m * d_m) * (-0.125d0)) * (((m_m * m_m) / d) * sqrt((l * h)))) / (l * l)
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) / Math.sqrt((l / d));
	} else if (d <= 6.5e-176) {
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * Math.sqrt((l * h)))) / (l * l);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) / math.sqrt((l / d))
	elif d <= 6.5e-176:
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * math.sqrt((l * h)))) / (l * l)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) / sqrt(Float64(l / d)));
	elseif (d <= 6.5e-176)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(Float64(M_m * M_m) / d) * sqrt(Float64(l * h)))) / Float64(l * l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (sqrt(-d) / sqrt(-h)) / sqrt((l / d));
	elseif (d <= 6.5e-176)
		tmp = (((D_m * D_m) * -0.125) * (((M_m * M_m) / d) * sqrt((l * h)))) / (l * l);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e-176], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 68.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.2

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

   5. Applied rewrites 11.2%

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

   7. Applied rewrites 45.2%

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

   9. Applied rewrites 50.4%

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

   if -4.999999999999985e-310 < d < 6.5e-176

   1. Initial program 23.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in d around 0

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

   8. Applied rewrites 43.0%

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

   if 6.5e-176 < d

   1. Initial program 73.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 68.2%

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

Alternative 16: 26.5% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

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

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 31.3

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

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
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 2024324 
(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)))))