Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 81.0%

Time: 15.9s

Alternatives: 17

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \mathsf{fma}\left(\frac{\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+120}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(t\_0 \cdot \sqrt{{\left(-\ell\right)}^{-1}}\right)\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-307}:\\ \;\;\;\;\left(t\_1 \cdot \frac{t\_0}{\sqrt{-h}}\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+212}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (sqrt (- d)))
        (t_1
         (fma
          (/ (* (* D_m (/ M_m d)) -0.5) l)
          (* (* (/ M_m d) h) (* 0.25 D_m))
          1.0))
        (t_2 (sqrt (/ d l))))
   (if (<= l -1.15e+120)
     (* (* t_1 (sqrt (/ d h))) (* t_0 (sqrt (pow (- l) -1.0))))
     (if (<= l -1.15e-307)
       (* (* t_1 (/ t_0 (sqrt (- h)))) t_2)
       (if (<= l 6.6e+212)
         (* (* t_1 (/ (sqrt d) (sqrt h))) t_2)
         (/ 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 = sqrt(-d);
	double t_1 = fma((((D_m * (M_m / d)) * -0.5) / l), (((M_m / d) * h) * (0.25 * D_m)), 1.0);
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -1.15e+120) {
		tmp = (t_1 * sqrt((d / h))) * (t_0 * sqrt(pow(-l, -1.0)));
	} else if (l <= -1.15e-307) {
		tmp = (t_1 * (t_0 / sqrt(-h))) * t_2;
	} else if (l <= 6.6e+212) {
		tmp = (t_1 * (sqrt(d) / sqrt(h))) * t_2;
	} 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 = sqrt(Float64(-d))
	t_1 = fma(Float64(Float64(Float64(D_m * Float64(M_m / d)) * -0.5) / l), Float64(Float64(Float64(M_m / d) * h) * Float64(0.25 * D_m)), 1.0)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.15e+120)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * Float64(t_0 * sqrt((Float64(-l) ^ -1.0))));
	elseif (l <= -1.15e-307)
		tmp = Float64(Float64(t_1 * Float64(t_0 / sqrt(Float64(-h)))) * t_2);
	elseif (l <= 6.6e+212)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(h))) * t_2);
	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[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.15e+120], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[Power[(-l), -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.15e-307], N[(N[(t$95$1 * N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 6.6e+212], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.14999999999999996e120

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 47.7

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

   6. Applied rewrites 47.7%

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

   8. Applied rewrites 55.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. div-inv N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-neg.f64 68.3

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

   10. Applied rewrites 68.3%

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

   if -1.14999999999999996e120 < l < -1.1499999999999999e-307

   1. Initial program 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   8. Applied rewrites 70.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 81.7

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

   10. Applied rewrites 81.7%

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

   if -1.1499999999999999e-307 < l < 6.6e212

   1. Initial program 72.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

   8. Applied rewrites 73.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 6.6e212 < l

   1. Initial program 17.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 70.8%

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

Alternative 2: 78.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{-0.5 \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(t\_0 \cdot h\right) \cdot \frac{M\_m}{d}, 0.25 \cdot D\_m, 1\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25 \cdot \left(h \cdot \frac{M\_m}{d}\right), D\_m \cdot t\_0, 1\right) \cdot t\_3\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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 (/ (* -0.5 (* (/ M_m d) D_m)) l))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (sqrt (/ d h)))
        (t_3 (sqrt (/ d l))))
   (if (<= t_1 -1e-191)
     (* (* (fma (* (* t_0 h) (/ M_m d)) (* 0.25 D_m) 1.0) t_2) t_3)
     (if (<= t_1 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_1 4e+148)
         (* (* (fma (* 0.25 (* h (/ M_m d))) (* D_m t_0) 1.0) t_3) t_2)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (-0.5 * ((M_m / d) * D_m)) / l;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / h));
	double t_3 = sqrt((d / l));
	double tmp;
	if (t_1 <= -1e-191) {
		tmp = (fma(((t_0 * h) * (M_m / d)), (0.25 * D_m), 1.0) * t_2) * t_3;
	} else if (t_1 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_1 <= 4e+148) {
		tmp = (fma((0.25 * (h * (M_m / d))), (D_m * t_0), 1.0) * t_3) * t_2;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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(-0.5 * Float64(Float64(M_m / d) * D_m)) / l)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / h))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -1e-191)
		tmp = Float64(Float64(fma(Float64(Float64(t_0 * h) * Float64(M_m / d)), Float64(0.25 * D_m), 1.0) * t_2) * t_3);
	elseif (t_1 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_1 <= 4e+148)
		tmp = Float64(Float64(fma(Float64(0.25 * Float64(h * Float64(M_m / d))), Float64(D_m * t_0), 1.0) * t_3) * t_2);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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[(-0.5 * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-191], N[(N[(N[(N[(N[(t$95$0 * h), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 4e+148], N[(N[(N[(N[(0.25 * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-191

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 79.7

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

   6. Applied rewrites 79.7%

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

   8. Applied rewrites 78.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 82.5%

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

   if -1e-191 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 77.6%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

   8. Applied rewrites 96.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   10. Applied rewrites 96.0%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{-0.5 \cdot \left(\frac{M}{d} \cdot D\right)}{\ell} \cdot h\right) \cdot \frac{M}{d}, 0.25 \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25 \cdot \left(h \cdot \frac{M}{d}\right), D \cdot \frac{-0.5 \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left(\mathsf{fma}\left(0.25 \cdot \left(h \cdot \frac{M\_m}{d}\right), D\_m \cdot \frac{-0.5 \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1
         (*
          (*
           (fma
            (* 0.25 (* h (/ M_m d)))
            (* D_m (/ (* -0.5 (* (/ M_m d) D_m)) l))
            1.0)
           (sqrt (/ d l)))
          (sqrt (/ d h)))))
   (if (<= t_0 -1e-191)
     t_1
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148) t_1 (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = (fma((0.25 * (h * (M_m / d))), (D_m * ((-0.5 * ((M_m / d) * D_m)) / l)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-191) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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((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))))
	t_1 = Float64(Float64(fma(Float64(0.25 * Float64(h * Float64(M_m / d))), Float64(D_m * Float64(Float64(-0.5 * Float64(Float64(M_m / d) * D_m)) / l)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -1e-191)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[(N[(N[(N[(0.25 * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(N[(-0.5 * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-191], t$95$1, If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], t$95$1, N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-191 or 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

   8. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   10. Applied rewrites 84.0%

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

   if -1e-191 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 77.6%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25 \cdot \left(h \cdot \frac{M}{d}\right), D \cdot \frac{-0.5 \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25 \cdot \left(h \cdot \frac{M}{d}\right), D \cdot \frac{-0.5 \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d \cdot \ell}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -5e-134)
     (*
      (*
       (fma
        (/ (* (* M_m D_m) -0.5) (* d l))
        (* (* (/ M_m d) h) (* 0.25 D_m))
        1.0)
       t_1)
      t_2)
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* t_2 t_1)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-134) {
		tmp = (fma((((M_m * D_m) * -0.5) / (d * l)), (((M_m / d) * h) * (0.25 * D_m)), 1.0) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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((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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -5e-134)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / Float64(d * l)), Float64(Float64(Float64(M_m / d) * h) * Float64(0.25 * D_m)), 1.0) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-134], N[(N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(t$95$2 * t$95$1), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-134

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

   8. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if -5.0000000000000003e-134 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 29.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 74.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d \cdot \ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(M\_m, \left(-0.125 \cdot M\_m\right) \cdot \left(\frac{D\_m \cdot D\_m}{\ell} \cdot h\right), d \cdot d\right)}{d \cdot d} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -1e-96)
     (*
      (*
       (/ (fma M_m (* (* -0.125 M_m) (* (/ (* D_m D_m) l) h)) (* d d)) (* d d))
       t_1)
      t_2)
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* t_2 t_1)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = ((fma(M_m, ((-0.125 * M_m) * (((D_m * D_m) / l) * h)), (d * d)) / (d * d)) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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((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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -1e-96)
		tmp = Float64(Float64(Float64(fma(M_m, Float64(Float64(-0.125 * M_m) * Float64(Float64(Float64(D_m * D_m) / l) * h)), Float64(d * d)) / Float64(d * d)) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-96], N[(N[(N[(N[(M$95$m * N[(N[(-0.125 * M$95$m), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(t$95$2 * t$95$1), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999991e-97

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   9. Applied rewrites 57.1%

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

   if -9.9999999999999991e-97 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 68.7%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(M, \left(-0.125 \cdot M\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot h\right), d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\left(\left(\left(\left(h \cdot \frac{M\_m \cdot M\_m}{\ell}\right) \cdot -0.125\right) \cdot D\_m\right) \cdot \frac{\frac{D\_m}{d}}{d}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -1e-96)
     (*
      (* (* (* (* (* h (/ (* M_m M_m) l)) -0.125) D_m) (/ (/ D_m d) d)) t_1)
      t_2)
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* t_2 t_1)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = (((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * ((D_m / d) / d)) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-1d-96)) then
        tmp = (((((h * ((m_m * m_m) / l)) * (-0.125d0)) * d_m) * ((d_m / d) / d)) * t_1) * t_2
    else if (t_0 <= 0.0d0) then
        tmp = abs((d / sqrt((l * h))))
    else if (t_0 <= 4d+148) then
        tmp = t_2 * t_1
    else
        tmp = (sqrt((h * l)) / abs(d)) ** (-1.0d0)
    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.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)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = (((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * ((D_m / d) / d)) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.pow((Math.sqrt((h * l)) / Math.abs(d)), -1.0);
	}
	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.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)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -1e-96:
		tmp = (((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * ((D_m / d) / d)) * t_1) * t_2
	elif t_0 <= 0.0:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif t_0 <= 4e+148:
		tmp = t_2 * t_1
	else:
		tmp = math.pow((math.sqrt((h * l)) / math.fabs(d)), -1.0)
	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((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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -1e-96)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(h * Float64(Float64(M_m * M_m) / l)) * -0.125) * D_m) * Float64(Float64(D_m / d) / d)) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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 = (((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)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -1e-96)
		tmp = (((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * ((D_m / d) / d)) * t_1) * t_2;
	elseif (t_0 <= 0.0)
		tmp = abs((d / sqrt((l * h))));
	elseif (t_0 <= 4e+148)
		tmp = t_2 * t_1;
	else
		tmp = (sqrt((h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-96], N[(N[(N[(N[(N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(t$95$2 * t$95$1), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999991e-97

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 60.8%

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

   if -9.9999999999999991e-97 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 68.7%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\left(\left(\left(\left(h \cdot \frac{M \cdot M}{\ell}\right) \cdot -0.125\right) \cdot D\right) \cdot \frac{\frac{D}{d}}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\left(h \cdot \left(\left(-0.125 \cdot \frac{M\_m \cdot M\_m}{\ell}\right) \cdot \left(\frac{\frac{D\_m}{d}}{d} \cdot D\_m\right)\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -1e-96)
     (*
      (* (* h (* (* -0.125 (/ (* M_m M_m) l)) (* (/ (/ D_m d) d) D_m))) t_1)
      t_2)
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* t_2 t_1)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = ((h * ((-0.125 * ((M_m * M_m) / l)) * (((D_m / d) / d) * D_m))) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-1d-96)) then
        tmp = ((h * (((-0.125d0) * ((m_m * m_m) / l)) * (((d_m / d) / d) * d_m))) * t_1) * t_2
    else if (t_0 <= 0.0d0) then
        tmp = abs((d / sqrt((l * h))))
    else if (t_0 <= 4d+148) then
        tmp = t_2 * t_1
    else
        tmp = (sqrt((h * l)) / abs(d)) ** (-1.0d0)
    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.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)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = ((h * ((-0.125 * ((M_m * M_m) / l)) * (((D_m / d) / d) * D_m))) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.pow((Math.sqrt((h * l)) / Math.abs(d)), -1.0);
	}
	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.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)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -1e-96:
		tmp = ((h * ((-0.125 * ((M_m * M_m) / l)) * (((D_m / d) / d) * D_m))) * t_1) * t_2
	elif t_0 <= 0.0:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif t_0 <= 4e+148:
		tmp = t_2 * t_1
	else:
		tmp = math.pow((math.sqrt((h * l)) / math.fabs(d)), -1.0)
	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((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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -1e-96)
		tmp = Float64(Float64(Float64(h * Float64(Float64(-0.125 * Float64(Float64(M_m * M_m) / l)) * Float64(Float64(Float64(D_m / d) / d) * D_m))) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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 = (((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)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -1e-96)
		tmp = ((h * ((-0.125 * ((M_m * M_m) / l)) * (((D_m / d) / d) * D_m))) * t_1) * t_2;
	elseif (t_0 <= 0.0)
		tmp = abs((d / sqrt((l * h))));
	elseif (t_0 <= 4e+148)
		tmp = t_2 * t_1;
	else
		tmp = (sqrt((h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-96], N[(N[(N[(h * N[(N[(-0.125 * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(t$95$2 * t$95$1), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999991e-97

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 60.8%

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

   12. Applied rewrites 59.7%

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

   if -9.9999999999999991e-97 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 68.7%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left(\left(h \cdot \left(\left(-0.125 \cdot \frac{M \cdot M}{\ell}\right) \cdot \left(\frac{\frac{D}{d}}{d} \cdot D\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{\left(\left(\left(h \cdot \frac{M\_m \cdot M\_m}{\ell}\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m}{d \cdot d} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -5e-134)
     (*
      (* (/ (* (* (* (* h (/ (* M_m M_m) l)) -0.125) D_m) D_m) (* d d)) t_1)
      t_2)
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* t_2 t_1)
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-134) {
		tmp = ((((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * D_m) / (d * d)) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-5d-134)) then
        tmp = ((((((h * ((m_m * m_m) / l)) * (-0.125d0)) * d_m) * d_m) / (d * d)) * t_1) * t_2
    else if (t_0 <= 0.0d0) then
        tmp = abs((d / sqrt((l * h))))
    else if (t_0 <= 4d+148) then
        tmp = t_2 * t_1
    else
        tmp = (sqrt((h * l)) / abs(d)) ** (-1.0d0)
    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.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)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-134) {
		tmp = ((((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * D_m) / (d * d)) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.pow((Math.sqrt((h * l)) / Math.abs(d)), -1.0);
	}
	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.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)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -5e-134:
		tmp = ((((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * D_m) / (d * d)) * t_1) * t_2
	elif t_0 <= 0.0:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif t_0 <= 4e+148:
		tmp = t_2 * t_1
	else:
		tmp = math.pow((math.sqrt((h * l)) / math.fabs(d)), -1.0)
	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((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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -5e-134)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(h * Float64(Float64(M_m * M_m) / l)) * -0.125) * D_m) * D_m) / Float64(d * d)) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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 = (((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)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -5e-134)
		tmp = ((((((h * ((M_m * M_m) / l)) * -0.125) * D_m) * D_m) / (d * d)) * t_1) * t_2;
	elseif (t_0 <= 0.0)
		tmp = abs((d / sqrt((l * h))));
	elseif (t_0 <= 4e+148)
		tmp = t_2 * t_1;
	else
		tmp = (sqrt((h * l)) / abs(d)) ^ -1.0;
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-134], N[(N[(N[(N[(N[(N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(t$95$2 * t$95$1), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-134

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 53.3

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 57.2%

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

   if -5.0000000000000003e-134 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 29.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 74.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{\left(\left(\left(h \cdot \frac{M \cdot M}{\ell}\right) \cdot -0.125\right) \cdot D\right) \cdot D}{d \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{-d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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))))))
   (if (<= t_0 -5e-134)
     (*
      (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 (- d)))
      (/ (sqrt (/ h l)) (fabs l)))
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 tmp;
	if (t_0 <= -5e-134) {
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / -d)) * (sqrt((h / l)) / fabs(l));
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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 = (((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)))
    if (t_0 <= (-5d-134)) then
        tmp = ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / -d)) * (sqrt((h / l)) / abs(l))
    else if (t_0 <= 0.0d0) then
        tmp = abs((d / sqrt((l * h))))
    else if (t_0 <= 4d+148) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (sqrt((h * l)) / abs(d)) ** (-1.0d0)
    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.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)));
	double tmp;
	if (t_0 <= -5e-134) {
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / -d)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.pow((Math.sqrt((h * l)) / Math.abs(d)), -1.0);
	}
	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.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)))
	tmp = 0
	if t_0 <= -5e-134:
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / -d)) * (math.sqrt((h / l)) / math.fabs(l))
	elif t_0 <= 0.0:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif t_0 <= 4e+148:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.pow((math.sqrt((h * l)) / math.fabs(d)), -1.0)
	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((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))))
	tmp = 0.0
	if (t_0 <= -5e-134)
		tmp = Float64(Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / Float64(-d))) * Float64(sqrt(Float64(h / l)) / abs(l)));
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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 = (((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)));
	tmp = 0.0;
	if (t_0 <= -5e-134)
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / -d)) * (sqrt((h / l)) / abs(l));
	elseif (t_0 <= 0.0)
		tmp = abs((d / sqrt((l * h))));
	elseif (t_0 <= 4e+148)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (sqrt((h * l)) / abs(d)) ^ -1.0;
	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[(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]}, If[LessEqual[t$95$0, -5e-134], N[(N[(N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / (-d)), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-134

   1. Initial program 79.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. 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. lower-*.f64 N/A

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

   8. Applied rewrites 30.6%

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

   10. Applied rewrites 37.2%

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

   if -5.0000000000000003e-134 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 29.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 74.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{-d}\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5000000000:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (*
          (* (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))))))
   (if (<= t_0 -5000000000.0)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= t_0 0.0)
       (fabs (/ d (sqrt (* l h))))
       (if (<= t_0 4e+148)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (pow (/ (sqrt (* h l)) (fabs d)) -1.0))))))

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 = (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 tmp;
	if (t_0 <= -5000000000.0) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (t_0 <= 0.0) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = pow((sqrt((h * l)) / fabs(d)), -1.0);
	}
	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 = (((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)))
    if (t_0 <= (-5000000000.0d0)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (t_0 <= 0.0d0) then
        tmp = abs((d / sqrt((l * h))))
    else if (t_0 <= 4d+148) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (sqrt((h * l)) / abs(d)) ** (-1.0d0)
    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.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)));
	double tmp;
	if (t_0 <= -5000000000.0) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (t_0 <= 4e+148) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.pow((Math.sqrt((h * l)) / Math.abs(d)), -1.0);
	}
	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.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)))
	tmp = 0
	if t_0 <= -5000000000.0:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif t_0 <= 0.0:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif t_0 <= 4e+148:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.pow((math.sqrt((h * l)) / math.fabs(d)), -1.0)
	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((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))))
	tmp = 0.0
	if (t_0 <= -5000000000.0)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (t_0 <= 0.0)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (t_0 <= 4e+148)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(h * l)) / abs(d)) ^ -1.0;
	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 = (((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)));
	tmp = 0.0;
	if (t_0 <= -5000000000.0)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (t_0 <= 0.0)
		tmp = abs((d / sqrt((l * h))));
	elseif (t_0 <= 4e+148)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (sqrt((h * l)) / abs(d)) ^ -1.0;
	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[(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]}, If[LessEqual[t$95$0, -5000000000.0], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+148], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e9

   1. Initial program 77.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 13.4

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

   5. Applied rewrites 13.4%

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

   if -5e9 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 49.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 54.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.0000000000000002e148

   1. Initial program 97.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 97.5%

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

   if 4.0000000000000002e148 < (*.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 26.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5000000000:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{h \cdot \ell}}{\left|d\right|}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5000000000:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

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))))
      -5000000000.0)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (fabs (/ 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) {
	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)))) <= -5000000000.0) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = fabs((d / sqrt((l * 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 / 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)))) <= (-5000000000.0d0)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = abs((d / sqrt((l * 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 (((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)))) <= -5000000000.0) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * 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 ((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)))) <= -5000000000.0:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.fabs((d / math.sqrt((l * 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 (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)))) <= -5000000000.0)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * 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 / 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)))) <= -5000000000.0)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = abs((d / sqrt((l * 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[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], -5000000000.0], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e9

   1. Initial program 77.5%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 13.4

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

   5. Applied rewrites 13.4%

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

   if -5e9 < (*.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 55.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 65.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5000000000:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

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))))
      -5e-134)
   (* (sqrt (pow (* l h) -1.0)) d)
   (fabs (/ 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) {
	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)))) <= -5e-134) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else {
		tmp = fabs((d / sqrt((l * 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 / 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)))) <= (-5d-134)) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else
        tmp = abs((d / sqrt((l * 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 (((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)))) <= -5e-134) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * 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 ((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)))) <= -5e-134:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	else:
		tmp = math.fabs((d / math.sqrt((l * 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 (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)))) <= -5e-134)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * 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 / 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)))) <= -5e-134)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	else
		tmp = abs((d / sqrt((l * 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[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], -5e-134], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-134

   1. Initial program 79.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 10.3

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

   5. Applied rewrites 10.3%

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

   if -5.0000000000000003e-134 < (*.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 52.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.4

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

   5. Applied rewrites 36.4%

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

   7. Applied rewrites 68.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

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 (sqrt (* l h)))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -5e-134)
     t_0
     (fabs t_0))))

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 / sqrt((l * h));
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-134) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	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 = d / sqrt((l * h))
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-134)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    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 = d / Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-134) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	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 = d / math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-134:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	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 / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-134)
		tmp = t_0;
	else
		tmp = abs(t_0);
	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 = d / sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-134)
		tmp = t_0;
	else
		tmp = abs(t_0);
	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[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-134], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-134

   1. Initial program 79.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 10.3

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

   5. Applied rewrites 10.3%

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

   7. Applied rewrites 9.3%

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

   if -5.0000000000000003e-134 < (*.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 52.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.4

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

   5. Applied rewrites 36.4%

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

   7. Applied rewrites 68.6%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\sqrt{-d} \cdot \sqrt{{\left(-\ell\right)}^{-1}}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

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
         (fma
          (/ (* (* D_m (/ M_m d)) -0.5) l)
          (* (* (/ M_m d) h) (* 0.25 D_m))
          1.0)))
   (if (<= d -5e-310)
     (* (* t_0 (sqrt (/ d h))) (* (sqrt (- d)) (sqrt (pow (- l) -1.0))))
     (if (<= d 1.75e-28)
       (*
        (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt d) (sqrt l)))
        (-
         1.0
         (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
       (* (* t_0 (/ (sqrt d) (sqrt h))) (sqrt (/ d 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 = fma((((D_m * (M_m / d)) * -0.5) / l), (((M_m / d) * h) * (0.25 * D_m)), 1.0);
	double tmp;
	if (d <= -5e-310) {
		tmp = (t_0 * sqrt((d / h))) * (sqrt(-d) * sqrt(pow(-l, -1.0)));
	} else if (d <= 1.75e-28) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(d) / sqrt(l))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * sqrt((d / 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 = fma(Float64(Float64(Float64(D_m * Float64(M_m / d)) * -0.5) / l), Float64(Float64(Float64(M_m / d) * h) * Float64(0.25 * D_m)), 1.0)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) * sqrt((Float64(-l) ^ -1.0))));
	elseif (d <= 1.75e-28)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * sqrt(Float64(d / 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[(N[(N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[Power[(-l), -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-28], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(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], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

   8. Applied rewrites 65.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. div-inv N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-neg.f64 73.2

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

   10. Applied rewrites 73.2%

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

   if -4.999999999999985e-310 < d < 1.75e-28

   1. Initial program 58.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 80.8

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

   4. Applied rewrites 80.8%

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

   if 1.75e-28 < d

   1. Initial program 67.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 67.8

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

   6. Applied rewrites 67.8%

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

   8. Applied rewrites 78.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 90.4

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

   10. Applied rewrites 90.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\sqrt{-d} \cdot \sqrt{{\left(-\ell\right)}^{-1}}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \mathsf{fma}\left(\frac{\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+120}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{t\_2}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-307}:\\ \;\;\;\;\left(t\_1 \cdot \frac{t\_2}{\sqrt{-h}}\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+212}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (sqrt (/ d l)))
        (t_1
         (fma
          (/ (* (* D_m (/ M_m d)) -0.5) l)
          (* (* (/ M_m d) h) (* 0.25 D_m))
          1.0))
        (t_2 (sqrt (- d))))
   (if (<= l -1.15e+120)
     (* (* t_1 (sqrt (/ d h))) (/ t_2 (sqrt (- l))))
     (if (<= l -1.15e-307)
       (* (* t_1 (/ t_2 (sqrt (- h)))) t_0)
       (if (<= l 6.6e+212)
         (* (* t_1 (/ (sqrt d) (sqrt h))) t_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 t_0 = sqrt((d / l));
	double t_1 = fma((((D_m * (M_m / d)) * -0.5) / l), (((M_m / d) * h) * (0.25 * D_m)), 1.0);
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -1.15e+120) {
		tmp = (t_1 * sqrt((d / h))) * (t_2 / sqrt(-l));
	} else if (l <= -1.15e-307) {
		tmp = (t_1 * (t_2 / sqrt(-h))) * t_0;
	} else if (l <= 6.6e+212) {
		tmp = (t_1 * (sqrt(d) / sqrt(h))) * t_0;
	} 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 = sqrt(Float64(d / l))
	t_1 = fma(Float64(Float64(Float64(D_m * Float64(M_m / d)) * -0.5) / l), Float64(Float64(Float64(M_m / d) * h) * Float64(0.25 * D_m)), 1.0)
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.15e+120)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * Float64(t_2 / sqrt(Float64(-l))));
	elseif (l <= -1.15e-307)
		tmp = Float64(Float64(t_1 * Float64(t_2 / sqrt(Float64(-h)))) * t_0);
	elseif (l <= 6.6e+212)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(h))) * t_0);
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.15e+120], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.15e-307], N[(N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 6.6e+212], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.14999999999999996e120

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 47.7

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

   6. Applied rewrites 47.7%

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

   8. Applied rewrites 55.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 68.3

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

   10. Applied rewrites 68.3%

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

   if -1.14999999999999996e120 < l < -1.1499999999999999e-307

   1. Initial program 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   8. Applied rewrites 70.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 81.7

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

   10. Applied rewrites 81.7%

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

   if -1.1499999999999999e-307 < l < 6.6e212

   1. Initial program 72.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

   8. Applied rewrites 73.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 6.6e212 < l

   1. Initial program 17.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 70.8%

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

Alternative 16: 77.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot -0.5}{\ell}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ \mathbf{if}\;\ell \leq -1.36 \cdot 10^{-291}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+212}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
         (fma
          (/ (* (* D_m (/ M_m d)) -0.5) l)
          (* (* (/ M_m d) h) (* 0.25 D_m))
          1.0)))
   (if (<= l -1.36e-291)
     (* (* t_0 (sqrt (/ d h))) (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l 6.6e+212)
       (* (* t_0 (/ (sqrt d) (sqrt h))) (sqrt (/ d 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 = fma((((D_m * (M_m / d)) * -0.5) / l), (((M_m / d) * h) * (0.25 * D_m)), 1.0);
	double tmp;
	if (l <= -1.36e-291) {
		tmp = (t_0 * sqrt((d / h))) * (sqrt(-d) / sqrt(-l));
	} else if (l <= 6.6e+212) {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * sqrt((d / 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 = fma(Float64(Float64(Float64(D_m * Float64(M_m / d)) * -0.5) / l), Float64(Float64(Float64(M_m / d) * h) * Float64(0.25 * D_m)), 1.0)
	tmp = 0.0
	if (l <= -1.36e-291)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= 6.6e+212)
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * sqrt(Float64(d / 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[(N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -1.36e-291], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+212], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.36000000000000007e-291

   1. Initial program 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

   8. Applied rewrites 65.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 73.5

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

   10. Applied rewrites 73.5%

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

   if -1.36000000000000007e-291 < l < 6.6e212

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 71.4

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

   6. Applied rewrites 71.4%

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

   8. Applied rewrites 72.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 79.5

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

   10. Applied rewrites 79.5%

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

   if 6.6e212 < l

   1. Initial program 17.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 70.8%

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

Alternative 17: 26.2% 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 62.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 26.9

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

5. Applied rewrites 26.9%

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

7. Applied rewrites 26.8%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(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)))))