Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 78.8%

Time: 22.3s

Alternatives: 24

Speedup: 3.3×

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: 66.1% 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: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{M\_m \cdot D}{d \cdot 2}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{-193}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}}\right) \cdot \left(1 + \frac{\frac{M\_m \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{t\_1}{\frac{-1}{h}}\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125, \left(D \cdot \left(\frac{h}{\ell} \cdot t\_0\right)\right) \cdot \left(M\_m \cdot \left(M\_m \cdot D\right)\right), t\_0 \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + \frac{t\_1}{\frac{1}{h}} \cdot \left(\left(\frac{M\_m \cdot D}{d} \cdot 0.25\right) \cdot \frac{-1}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))) (t_1 (/ (* M_m D) (* d 2.0))))
   (if (<= d -8.5e-193)
     (*
      (* (pow (/ d h) (/ 1.0 2.0)) (/ 1.0 (/ (sqrt (- l)) (sqrt (- d)))))
      (+ 1.0 (* (/ (/ (* M_m (* D 0.5)) (* d 2.0)) l) (/ t_1 (/ -1.0 h)))))
     (if (<= d 1.8e-276)
       (/
        (/
         (fma
          -0.125
          (* (* D (* (/ h l) t_0)) (* M_m (* M_m D)))
          (* t_0 (* d d)))
         d)
        h)
       (*
        (* (* (sqrt d) (sqrt (/ 1.0 h))) (/ 1.0 (sqrt (/ l d))))
        (+
         1.0
         (* (/ t_1 (/ 1.0 h)) (* (* (/ (* M_m D) d) 0.25) (/ -1.0 l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (M_m * D) / (d * 2.0);
	double tmp;
	if (d <= -8.5e-193) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (1.0 / (sqrt(-l) / sqrt(-d)))) * (1.0 + ((((M_m * (D * 0.5)) / (d * 2.0)) / l) * (t_1 / (-1.0 / h))));
	} else if (d <= 1.8e-276) {
		tmp = (fma(-0.125, ((D * ((h / l) * t_0)) * (M_m * (M_m * D))), (t_0 * (d * d))) / d) / h;
	} else {
		tmp = ((sqrt(d) * sqrt((1.0 / h))) * (1.0 / sqrt((l / d)))) * (1.0 + ((t_1 / (1.0 / h)) * ((((M_m * D) / d) * 0.25) * (-1.0 / l))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64(M_m * D) / Float64(d * 2.0))
	tmp = 0.0
	if (d <= -8.5e-193)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(1.0 / Float64(sqrt(Float64(-l)) / sqrt(Float64(-d))))) * Float64(1.0 + Float64(Float64(Float64(Float64(M_m * Float64(D * 0.5)) / Float64(d * 2.0)) / l) * Float64(t_1 / Float64(-1.0 / h)))));
	elseif (d <= 1.8e-276)
		tmp = Float64(Float64(fma(-0.125, Float64(Float64(D * Float64(Float64(h / l) * t_0)) * Float64(M_m * Float64(M_m * D))), Float64(t_0 * Float64(d * d))) / d) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(Float64(1.0 / h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 + Float64(Float64(t_1 / Float64(1.0 / h)) * Float64(Float64(Float64(Float64(M_m * D) / d) * 0.25) * Float64(-1.0 / l)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.5e-193], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[(-l)], $MachinePrecision] / N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(M$95$m * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-276], N[(N[(N[(-0.125 * N[(N[(D * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(t$95$1 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.50000000000000004e-193

   1. Initial program 68.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 72.4

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

   6. Applied rewrites 72.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 87.3

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

   8. Applied rewrites 87.3%

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

   if -8.50000000000000004e-193 < d < 1.79999999999999997e-276

   1. Initial program 41.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in d around 0

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

   8. Applied rewrites 25.4%

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

   10. Applied rewrites 51.9%

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

   12. Applied rewrites 69.5%

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

   if 1.79999999999999997e-276 < d

   1. Initial program 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.6

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 68.6

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

   8. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. pow-pow N/A

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

      21. inv-pow N/A

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

      22. lift-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. pow1/2 N/A

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

      25. lower-sqrt.f64 81.2

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

   10. Applied rewrites 81.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-193}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}}\right) \cdot \left(1 + \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125, \left(D \cdot \left(\frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}\right)\right) \cdot \left(M \cdot \left(M \cdot D\right)\right), \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{h}} \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \frac{-1}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-160}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M\_m \cdot D\right) \cdot \frac{\left(M\_m \cdot D\right) \cdot 0.25}{d}}{\ell \cdot \left(d \cdot 2\right)}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot \left(M\_m \cdot \left(D \cdot \left(M\_m \cdot D\right)\right)\right)}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_0 -5e-160)
     (*
      (- 1.0 (* h (/ (* (* M_m D) (/ (* (* M_m D) 0.25) d)) (* l (* d 2.0)))))
      (sqrt (* d (/ d (* h l)))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (/
        (*
         d
         (+
          1.0
          (/ (* (* h -0.5) (* M_m (* D (* M_m D)))) (* l (* (* d d) 4.0)))))
        (sqrt (* h l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-160) {
		tmp = (1.0 - (h * (((M_m * D) * (((M_m * D) * 0.25) / d)) / (l * (d * 2.0))))) * sqrt((d * (d / (h * l))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = (d * (1.0 + (((h * -0.5) * (M_m * (D * (M_m * D)))) / (l * ((d * d) * 4.0))))) / sqrt((h * l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-160) {
		tmp = (1.0 - (h * (((M_m * D) * (((M_m * D) * 0.25) / d)) / (l * (d * 2.0))))) * Math.sqrt((d * (d / (h * l))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = (d * (1.0 + (((h * -0.5) * (M_m * (D * (M_m * D)))) / (l * ((d * d) * 4.0))))) / Math.sqrt((h * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_0 <= -5e-160:
		tmp = (1.0 - (h * (((M_m * D) * (((M_m * D) * 0.25) / d)) / (l * (d * 2.0))))) * math.sqrt((d * (d / (h * l))))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = (d * (1.0 + (((h * -0.5) * (M_m * (D * (M_m * D)))) / (l * ((d * d) * 4.0))))) / math.sqrt((h * l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -5e-160)
		tmp = Float64(Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D) * Float64(Float64(Float64(M_m * D) * 0.25) / d)) / Float64(l * Float64(d * 2.0))))) * sqrt(Float64(d * Float64(d / Float64(h * l)))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(d * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * Float64(M_m * Float64(D * Float64(M_m * D)))) / Float64(l * Float64(Float64(d * d) * 4.0))))) / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -5e-160)
		tmp = (1.0 - (h * (((M_m * D) * (((M_m * D) * 0.25) / d)) / (l * (d * 2.0))))) * sqrt((d * (d / (h * l))));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = (d * (1.0 + (((h * -0.5) * (M_m * (D * (M_m * D)))) / (l * ((d * d) * 4.0))))) / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-160], N[(N[(1.0 - N[(h * N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(N[(N[(M$95$m * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[(M$95$m * N[(D * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < -4.99999999999999994e-160

   1. Initial program 86.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 90.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 89.5

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

   6. Applied rewrites 89.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.5

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 89.5

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

   8. Applied rewrites 89.5%

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

   10. Applied rewrites 68.2%

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

   if -4.99999999999999994e-160 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 77.3%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 74.8%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{0.25 \cdot \left(\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot h\right) \cdot \left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4} \cdot h\right)\right)}{\ell \cdot \ell}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, 0.5 \cdot \frac{h}{\ell}, 1\right)}} \]

   6. Applied rewrites 26.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -5 \cdot 10^{-160}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \frac{\left(M \cdot D\right) \cdot 0.25}{d}}{\ell \cdot \left(d \cdot 2\right)}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(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)))))