Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 83.0%

Time: 16.9s

Alternatives: 17

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\_m\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \frac{M\_m}{d} \cdot h\\ t_3 := \mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot t\_0}{\ell}, \left(0.25 \cdot D\_m\right) \cdot t\_2, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.25 \cdot \left(-D\_m\right)\right) \cdot \frac{t\_0 \cdot D\_m}{\ell}, t\_2, 1\right) \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 d) M_m))
        (t_1
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2 (* (/ M_m d) h))
        (t_3
         (*
          (fma (/ (* (- D_m) t_0) l) (* (* 0.25 D_m) t_2) 1.0)
          (/ (fabs d) (sqrt (* l h))))))
   (if (<= t_1 0.0)
     t_3
     (if (<= t_1 5e+132)
       (*
        (fma (* (* 0.25 (- D_m)) (/ (* t_0 D_m) l)) t_2 1.0)
        (/ (sqrt (/ d l)) (sqrt (/ h d))))
       t_3))))

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 / d) * M_m;
	double t_1 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = (M_m / d) * h;
	double t_3 = fma(((-D_m * t_0) / l), ((0.25 * D_m) * t_2), 1.0) * (fabs(d) / sqrt((l * h)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_3;
	} else if (t_1 <= 5e+132) {
		tmp = fma(((0.25 * -D_m) * ((t_0 * D_m) / l)), t_2, 1.0) * (sqrt((d / l)) / sqrt((h / d)));
	} else {
		tmp = t_3;
	}
	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 / d) * M_m)
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = Float64(Float64(M_m / d) * h)
	t_3 = Float64(fma(Float64(Float64(Float64(-D_m) * t_0) / l), Float64(Float64(0.25 * D_m) * t_2), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_3;
	elseif (t_1 <= 5e+132)
		tmp = Float64(fma(Float64(Float64(0.25 * Float64(-D_m)) * Float64(Float64(t_0 * D_m) / l)), t_2, 1.0) * Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d))));
	else
		tmp = t_3;
	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 / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[((-D$95$m) * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$3, If[LessEqual[t$95$1, 5e+132], N[(N[(N[(N[(0.25 * (-D$95$m)), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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)))) < 0.0 or 5.0000000000000001e132 < (*.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 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 64.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. rem-sqrt-square N/A

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

      21. lower-fabs.f64 81.7

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 81.7

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

   10. Applied rewrites 81.7%

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

   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)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

   8. Applied rewrites 97.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.25 \cdot \left(-D\right)\right) \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}, \frac{M}{d} \cdot h, 1\right) \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \frac{\left|d\right|}{t\_1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(\frac{\frac{M\_m \cdot M\_m}{d}}{d} \cdot h\right) \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot -0.125}{\ell}\right) \cdot \frac{d}{t\_1}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (* l h)))
        (t_2 (/ (fabs d) t_1)))
   (if (<= t_0 -2e-45)
     (*
      (* (* (/ (/ (* M_m M_m) d) d) h) (/ (* (* D_m D_m) -0.125) l))
      (/ d t_1))
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+132) (* (sqrt (/ d h)) (sqrt (/ d l))) t_2)))))

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 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((l * h));
	double t_2 = fabs(d) / t_1;
	double tmp;
	if (t_0 <= -2e-45) {
		tmp = (((((M_m * M_m) / d) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (d / t_1);
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_2;
	}
	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 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = sqrt((l * h))
    t_2 = abs(d) / t_1
    if (t_0 <= (-2d-45)) then
        tmp = (((((m_m * m_m) / d) / d) * h) * (((d_m * d_m) * (-0.125d0)) / l)) * (d / t_1)
    else if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 5d+132) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_2
    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 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = Math.sqrt((l * h));
	double t_2 = Math.abs(d) / t_1;
	double tmp;
	if (t_0 <= -2e-45) {
		tmp = (((((M_m * M_m) / d) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (d / t_1);
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+132) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_2;
	}
	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 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = math.sqrt((l * h))
	t_2 = math.fabs(d) / t_1
	tmp = 0
	if t_0 <= -2e-45:
		tmp = (((((M_m * M_m) / d) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (d / t_1)
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 5e+132:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_2
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(abs(d) / t_1)
	tmp = 0.0
	if (t_0 <= -2e-45)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) * h) * Float64(Float64(Float64(D_m * D_m) * -0.125) / l)) * Float64(d / t_1));
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_2;
	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 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = sqrt((l * h));
	t_2 = abs(d) / t_1;
	tmp = 0.0;
	if (t_0 <= -2e-45)
		tmp = (((((M_m * M_m) / d) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (d / t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+132)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_2;
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-45], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999997e-45

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 88.2%

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

   8. Applied rewrites 37.5%

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

   9. Taylor expanded in h around inf

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*l/ N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 29.8

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

   11. Applied rewrites 29.8%

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

   if -1.99999999999999997e-45 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000001e132 < (*.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 33.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.7

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

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 23.6%

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

   12. Applied rewrites 62.4%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{\ell}{d} \cdot h}{-d}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 -5e+206)
     (/ 1.0 (sqrt (/ (* (/ l d) h) (- d))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 5e+132) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

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 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= -5e+206) {
		tmp = 1.0 / sqrt((((l / d) * h) / -d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = abs(d) / sqrt((l * h))
    if (t_0 <= (-5d+206)) then
        tmp = 1.0d0 / sqrt((((l / d) * h) / -d))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+132) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_1
    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 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double tmp;
	if (t_0 <= -5e+206) {
		tmp = 1.0 / Math.sqrt((((l / d) * h) / -d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+132) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_1;
	}
	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 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	tmp = 0
	if t_0 <= -5e+206:
		tmp = 1.0 / math.sqrt((((l / d) * h) / -d))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+132:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_1
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -5e+206)
		tmp = Float64(1.0 / sqrt(Float64(Float64(Float64(l / d) * h) / Float64(-d))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	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 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = abs(d) / sqrt((l * h));
	tmp = 0.0;
	if (t_0 <= -5e+206)
		tmp = 1.0 / sqrt((((l / d) * h) / -d));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+132)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_1;
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+206], N[(1.0 / N[Sqrt[N[(N[(N[(l / d), $MachinePrecision] * h), $MachinePrecision] / (-d)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000002e206

   1. Initial program 78.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 3.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-2neg N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. frac-times N/A

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

      16. sqrt-div N/A

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

   7. Applied rewrites 7.2%

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

   9. Applied rewrites 18.1%

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

   if -5.0000000000000002e206 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000001e132 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 38.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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

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

   8. Applied rewrites 28.7%

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

   10. Applied rewrites 21.8%

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

   12. Applied rewrites 57.7%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{\ell}{d} \cdot h}{-d}}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (/ (fabs d) (sqrt (* l h))))
        (t_1
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -2e-45)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= t_1 1e-154)
       t_0
       (if (<= t_1 5e+132) (* (sqrt (* (/ (/ d h) l) d)) 1.0) 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 = fabs(d) / sqrt((l * h));
	double t_1 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-45) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 1e-154) {
		tmp = t_0;
	} else if (t_1 <= 5e+132) {
		tmp = sqrt((((d / h) / l) * d)) * 1.0;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = abs(d) / sqrt((l * h))
    t_1 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-2d-45)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else if (t_1 <= 1d-154) then
        tmp = t_0
    else if (t_1 <= 5d+132) then
        tmp = sqrt((((d / h) / l) * d)) * 1.0d0
    else
        tmp = 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 = Math.abs(d) / Math.sqrt((l * h));
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-45) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else if (t_1 <= 1e-154) {
		tmp = t_0;
	} else if (t_1 <= 5e+132) {
		tmp = Math.sqrt((((d / h) / l) * d)) * 1.0;
	} else {
		tmp = 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 = math.fabs(d) / math.sqrt((l * h))
	t_1 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -2e-45:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	elif t_1 <= 1e-154:
		tmp = t_0
	elif t_1 <= 5e+132:
		tmp = math.sqrt((((d / h) / l) * d)) * 1.0
	else:
		tmp = 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(abs(d) / sqrt(Float64(l * h)))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -2e-45)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (t_1 <= 1e-154)
		tmp = t_0;
	elseif (t_1 <= 5e+132)
		tmp = Float64(sqrt(Float64(Float64(Float64(d / h) / l) * d)) * 1.0);
	else
		tmp = 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 = abs(d) / sqrt((l * h));
	t_1 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -2e-45)
		tmp = sqrt((1.0 / (l * h))) * -d;
	elseif (t_1 <= 1e-154)
		tmp = t_0;
	elseif (t_1 <= 5e+132)
		tmp = sqrt((((d / h) / l) * d)) * 1.0;
	else
		tmp = 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[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-45], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 1e-154], t$95$0, If[LessEqual[t$95$1, 5e+132], N[(N[Sqrt[N[(N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999997e-45

   1. Initial program 80.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 12.8

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

   8. Applied rewrites 12.8%

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

   if -1.99999999999999997e-45 < (*.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)))) < 9.9999999999999997e-155 or 5.0000000000000001e132 < (*.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 37.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.8

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

   8. Applied rewrites 30.8%

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

   10. Applied rewrites 23.1%

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

   12. Applied rewrites 61.4%

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

   if 9.9999999999999997e-155 < (*.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)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 95.0%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-154}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right)}{\ell}, \left(0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1
         (*
          (fma
           (/ (* (- D_m) (* (/ 0.5 d) M_m)) l)
           (* (* 0.25 D_m) (* (/ M_m d) h))
           1.0)
          (/ (fabs d) (sqrt (* l h))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+132) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1))))

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 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = fma(((-D_m * ((0.5 / d) * M_m)) / l), ((0.25 * D_m) * ((M_m / d) * h)), 1.0) * (fabs(d) / sqrt((l * h)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(fma(Float64(Float64(Float64(-D_m) * Float64(Float64(0.5 / d) * M_m)) / l), Float64(Float64(0.25 * D_m) * Float64(Float64(M_m / d) * h)), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[((-D$95$m) * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)))) < 0.0 or 5.0000000000000001e132 < (*.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 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 64.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. rem-sqrt-square N/A

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

      21. lower-fabs.f64 81.7

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 81.7

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

   10. Applied rewrites 81.7%

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

   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)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{M\_m}{d} \cdot D\_m\\ t_2 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{d}{t\_0} \cdot \mathsf{fma}\left(t\_1, \left(-0.125 \cdot \frac{h}{\ell}\right) \cdot t\_1, 1\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \end{array} \end{array} \]

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 (* l h)))
        (t_1 (* (/ M_m d) D_m))
        (t_2
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_2 0.0)
     (* (/ d t_0) (fma t_1 (* (* -0.125 (/ h l)) t_1) 1.0))
     (if (<= t_2 5e+132) (* (sqrt (/ d h)) (sqrt (/ d l))) (/ (fabs d) 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 = sqrt((l * h));
	double t_1 = (M_m / d) * D_m;
	double t_2 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (d / t_0) * fma(t_1, ((-0.125 * (h / l)) * t_1), 1.0);
	} else if (t_2 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(d) / 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 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(M_m / d) * D_m)
	t_2 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(d / t_0) * fma(t_1, Float64(Float64(-0.125 * Float64(h / l)) * t_1), 1.0));
	elseif (t_2 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(abs(d) / t_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[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(d / t$95$0), $MachinePrecision] * N[(t$95$1 * N[(N[(-0.125 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 83.5%

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

   8. Applied rewrites 40.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.4

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

   10. Applied rewrites 42.4%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 23.8%

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

   12. Applied rewrites 58.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M}{d} \cdot D, \left(-0.125 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D\_m}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot D\_m}{d}\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{t\_1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_1}\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (* l h))))
   (if (<= t_0 0.0)
     (*
      (fma (* (* (/ D_m d) (/ (* (* M_m M_m) D_m) d)) -0.125) (/ h l) 1.0)
      (/ d t_1))
     (if (<= t_0 5e+132) (* (sqrt (/ d h)) (sqrt (/ d l))) (/ (fabs d) t_1)))))

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 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma((((D_m / d) * (((M_m * M_m) * D_m) / d)) * -0.125), (h / l), 1.0) * (d / t_1);
	} else if (t_0 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(d) / t_1;
	}
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(l * h))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(D_m / d) * Float64(Float64(Float64(M_m * M_m) * D_m) / d)) * -0.125), Float64(h / l), 1.0) * Float64(d / t_1));
	elseif (t_0 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(abs(d) / t_1);
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 83.5%

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

   8. Applied rewrites 40.9%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 39.4

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

   11. Applied rewrites 39.4%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 23.8%

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

   12. Applied rewrites 58.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{d} \cdot \frac{\left(M \cdot M\right) \cdot D}{d}\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_0 0.0)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= t_0 5e+132)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (/ (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 t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (t_0 <= 5e+132) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else if (t_0 <= 5d+132) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    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 t_0 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else if (t_0 <= 5e+132) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} 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):
	t_0 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	elif t_0 <= 5e+132:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	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)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (t_0 <= 5e+132)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(abs(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)
	t_0 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((1.0 / (l * h))) * -d;
	elseif (t_0 <= 5e+132)
		tmp = sqrt((d / h)) * sqrt((d / l));
	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_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$0, 5e+132], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 18.1

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

   8. Applied rewrites 18.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e132

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 98.6%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 23.8%

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

   12. Applied rewrites 58.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\_m\\ \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(0.5 \cdot D\_m\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{t\_0 \cdot D\_m}{\ell}\right) \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot t\_0}{\ell}, \left(0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (* (/ 0.5 d) M_m)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        5e+132)
     (*
      (-
       1.0
       (*
        (/ (* (/ M_m d) (* (* 0.5 D_m) 0.5)) (pow h -1.0))
        (/ (* t_0 D_m) l)))
      (/ (sqrt (/ d l)) (sqrt (/ h d))))
     (*
      (fma (/ (* (- D_m) t_0) l) (* (* 0.25 D_m) (* (/ M_m d) 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 t_0 = (0.5 / d) * M_m;
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+132) {
		tmp = (1.0 - ((((M_m / d) * ((0.5 * D_m) * 0.5)) / pow(h, -1.0)) * ((t_0 * D_m) / l))) * (sqrt((d / l)) / sqrt((h / d)));
	} else {
		tmp = fma(((-D_m * t_0) / l), ((0.25 * D_m) * ((M_m / d) * h)), 1.0) * (fabs(d) / 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)
	t_0 = Float64(Float64(0.5 / d) * M_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+132)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(0.5 * D_m) * 0.5)) / (h ^ -1.0)) * Float64(Float64(t_0 * D_m) / l))) * Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(-D_m) * t_0) / l), Float64(Float64(0.25 * D_m) * Float64(Float64(M_m / d) * h)), 1.0) * Float64(abs(d) / sqrt(Float64(l * 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[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+132], N[(N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(0.5 * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-D$95$m) * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / 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)))) < 5.0000000000000001e132

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 89.3%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 43.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. rem-sqrt-square N/A

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

      21. lower-fabs.f64 78.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 78.2

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

   10. Applied rewrites 78.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}\right) \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left(-D\_m\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right)}{\ell}\\ \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot t\_0, h, 1\right)}{\sqrt{\frac{\ell}{d}} \cdot \sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (/ (* (- D_m) (* (/ 0.5 d) M_m)) l)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        5e+132)
     (/
      (fma (* (* (* 0.25 D_m) (/ M_m d)) t_0) h 1.0)
      (* (sqrt (/ l d)) (sqrt (/ h d))))
     (*
      (fma t_0 (* (* 0.25 D_m) (* (/ M_m d) 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 t_0 = (-D_m * ((0.5 / d) * M_m)) / l;
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+132) {
		tmp = fma((((0.25 * D_m) * (M_m / d)) * t_0), h, 1.0) / (sqrt((l / d)) * sqrt((h / d)));
	} else {
		tmp = fma(t_0, ((0.25 * D_m) * ((M_m / d) * h)), 1.0) * (fabs(d) / 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)
	t_0 = Float64(Float64(Float64(-D_m) * Float64(Float64(0.5 / d) * M_m)) / l)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+132)
		tmp = Float64(fma(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * t_0), h, 1.0) / Float64(sqrt(Float64(l / d)) * sqrt(Float64(h / d))));
	else
		tmp = Float64(fma(t_0, Float64(Float64(0.25 * D_m) * Float64(Float64(M_m / d) * h)), 1.0) * Float64(abs(d) / sqrt(Float64(l * 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[((-D$95$m) * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+132], N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * h + 1.0), $MachinePrecision] / N[(N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / 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)))) < 5.0000000000000001e132

   1. Initial program 84.9%

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

      1. lift-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. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 46.1

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

   4. Applied rewrites 46.1%

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

   6. Applied rewrites 62.9%

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

   8. Applied rewrites 65.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. sqrt-prod N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. associate-/r/ N/A

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

      15. pow1/2 N/A

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

      16. sqrt-div N/A

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

      17. lift-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   10. Applied rewrites 89.2%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 43.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. rem-sqrt-square N/A

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

      21. lower-fabs.f64 78.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 78.2

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

   10. Applied rewrites 78.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, h, 1\right)}{\sqrt{\frac{\ell}{d}} \cdot \sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right)}{\ell}, \left(0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;t\_0 \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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) (* (/ 0.5 d) M_m)) l)
          (* (* 0.25 D_m) (* (/ M_m d) h))
          1.0)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        5e+132)
     (* t_0 (/ (sqrt (/ d l)) (sqrt (/ h d))))
     (* t_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 t_0 = fma(((-D_m * ((0.5 / d) * M_m)) / l), ((0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+132) {
		tmp = t_0 * (sqrt((d / l)) / sqrt((h / d)));
	} else {
		tmp = t_0 * (fabs(d) / 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)
	t_0 = fma(Float64(Float64(Float64(-D_m) * Float64(Float64(0.5 / d) * M_m)) / l), Float64(Float64(0.25 * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+132)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d))));
	else
		tmp = Float64(t_0 * Float64(abs(d) / sqrt(Float64(l * 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[((-D$95$m) * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+132], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Abs[d], $MachinePrecision] / 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)))) < 5.0000000000000001e132

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 89.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 87.5%

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

   if 5.0000000000000001e132 < (*.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 32.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 43.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      8. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      14. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      17. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      20. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      21. lower-fabs.f64 78.2

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      23. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \]

      24. lower-*.f64 78.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \]

   10. Applied rewrites 78.2%

       \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\ell}, \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -2e-45)
   (* (sqrt (/ 1.0 (* l h))) (- 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 (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -2e-45) {
		tmp = sqrt((1.0 / (l * h))) * -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 (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-2d-45)) then
        tmp = sqrt((1.0d0 / (l * h))) * -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 (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -2e-45) {
		tmp = Math.sqrt((1.0 / (l * h))) * -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 ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -2e-45:
		tmp = math.sqrt((1.0 / (l * h))) * -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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -2e-45)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(abs(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 (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -2e-45)
		tmp = sqrt((1.0 / (l * h))) * -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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-45], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -1.99999999999999997e-45

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 3.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 12.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 12.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.99999999999999997e-45 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 34.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.6%

       \[\leadsto \frac{\sqrt{-d}}{\color{blue}{\sqrt{\frac{h}{d} \cdot \left(-\ell\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 64.4%

       \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -1e+57)
   (* (sqrt (/ 1.0 (* l h))) 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 (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e+57) {
		tmp = sqrt((1.0 / (l * h))) * 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 (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d+57)) then
        tmp = sqrt((1.0d0 / (l * h))) * 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 (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -1e+57) {
		tmp = Math.sqrt((1.0 / (l * h))) * 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 ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -1e+57:
		tmp = math.sqrt((1.0 / (l * h))) * 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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -1e+57)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	else
		tmp = Float64(abs(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 (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e+57)
		tmp = sqrt((1.0 / (l * h))) * 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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+57], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -1.00000000000000005e57

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 3.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 10.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if -1.00000000000000005e57 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 33.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.2%

       \[\leadsto \frac{\sqrt{-d}}{\color{blue}{\sqrt{\frac{h}{d} \cdot \left(-\ell\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \end{array} \end{array} \]

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 (* l h))))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        -1e+57)
     (/ d t_0)
     (/ (fabs d) 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 = sqrt((l * h));
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e+57) {
		tmp = d / t_0;
	} else {
		tmp = fabs(d) / 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 = sqrt((l * h))
    if (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d+57)) then
        tmp = d / t_0
    else
        tmp = abs(d) / 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 = Math.sqrt((l * h));
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -1e+57) {
		tmp = d / t_0;
	} else {
		tmp = Math.abs(d) / 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 = math.sqrt((l * h))
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -1e+57:
		tmp = d / t_0
	else:
		tmp = math.fabs(d) / 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 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -1e+57)
		tmp = Float64(d / t_0);
	else
		tmp = Float64(abs(d) / 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 = sqrt((l * h));
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e+57)
		tmp = d / t_0;
	else
		tmp = abs(d) / 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[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+57], N[(d / t$95$0), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / 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)))) < -1.00000000000000005e57

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 3.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 10.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   if -1.00000000000000005e57 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 33.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.2%

       \[\leadsto \frac{\sqrt{-d}}{\color{blue}{\sqrt{\frac{h}{d} \cdot \left(-\ell\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 3.3e-266)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ (/ d (sqrt h)) (sqrt l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3.3e-266) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	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 (l <= 3.3d-266) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    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 (l <= 3.3e-266) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	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 l <= 3.3e-266:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 3.3e-266)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	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 (l <= 3.3e-266)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	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[l, 3.3e-266], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3000000000000003e-266

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 40.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 3.3000000000000003e-266 < l

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 44.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 46.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.4%

       \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 3.3e-266)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3.3e-266) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 3.3d-266) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3.3e-266) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 3.3e-266:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 3.3e-266)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 3.3e-266)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 3.3e-266], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3000000000000003e-266

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 40.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 3.3000000000000003e-266 < l

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 44.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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 46.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.4%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.4% 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 65.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 39.0%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. 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 25.8

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 25.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 25.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(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)))))