Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 80.6%

Time: 20.1s

Alternatives: 26

Speedup: 3.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D \cdot 0.25\right)\\ t_1 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{t\_1}{\frac{1}{h}} \cdot \frac{\frac{t\_0}{d}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \frac{t\_0}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (* M_m (* D 0.25))) (t_1 (* (* (/ 0.5 d) D) M_m)))
   (if (<= d -5e-310)
     (*
      (- 1.0 (* (/ t_1 (/ 1.0 h)) (/ (/ t_0 d) l)))
      (* (/ (sqrt (- d)) (sqrt (- l))) (pow (/ d h) (/ 1.0 2.0))))
     (/ (* (fma (* t_1 (/ t_0 (* (- l) d))) h 1.0) d) (* (sqrt h) (sqrt 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 = M_m * (D * 0.25);
	double t_1 = ((0.5 / d) * D) * M_m;
	double tmp;
	if (d <= -5e-310) {
		tmp = (1.0 - ((t_1 / (1.0 / h)) * ((t_0 / d) / l))) * ((sqrt(-d) / sqrt(-l)) * pow((d / h), (1.0 / 2.0)));
	} else {
		tmp = (fma((t_1 * (t_0 / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(M_m * Float64(D * 0.25))
	t_1 = Float64(Float64(Float64(0.5 / d) * D) * M_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(t_1 / Float64(1.0 / h)) * Float64(Float64(t_0 / d) / l))) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	else
		tmp = Float64(Float64(fma(Float64(t_1 * Float64(t_0 / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(1.0 - N[(N[(t$95$1 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * N[(t$95$0 / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. 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 76.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 76.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 76.2

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

   6. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   8. Applied rewrites 76.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 86.0

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

   10. Applied rewrites 86.0%

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

   if -4.999999999999985e-310 < d

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      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 71.3%

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

   6. Applied rewrites 62.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 83.8%

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

Alternative 2: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M\_m \cdot D\right)}{\ell \cdot d}, \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot h, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 -1e-167)
     (*
      (*
       (fma (/ (* -0.25 (* M_m D)) (* l d)) (* (* (* (/ 0.5 d) D) M_m) h) 1.0)
       t_1)
      t_2)
     (if (<= t_0 0.0) t_3 (if (<= t_0 2e+264) (* t_2 t_1) t_3)))))

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 = (1.0 - ((h / l) * (pow(((M_m * D) / (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((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= -1e-167) {
		tmp = (fma(((-0.25 * (M_m * D)) / (l * d)), ((((0.5 / d) * D) * M_m) * h), 1.0) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 2e+264) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -1e-167)
		tmp = Float64(Float64(fma(Float64(Float64(-0.25 * Float64(M_m * D)) / Float64(l * d)), Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * h), 1.0) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 2e+264)
		tmp = Float64(t_2 * t_1);
	else
		tmp = t_3;
	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[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-167], N[(N[(N[(N[(N[(-0.25 * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$3, If[LessEqual[t$95$0, 2e+264], N[(t$95$2 * t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-167

   1. Initial program 88.7%

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

      1. lift-*.f64 N/A

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

      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 89.5%

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

   6. Applied rewrites 30.8%

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

   8. Applied rewrites 84.6%

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

   if -1e-167 < (*.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 2.00000000000000009e264 < (*.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 27.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 57.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell \cdot d}, \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M\_m \cdot D\right)}{\ell \cdot d} \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right), h, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 -1e-167)
     (*
      (*
       (fma (* (/ (* -0.25 (* M_m D)) (* l d)) (* (* (/ 0.5 d) D) M_m)) h 1.0)
       t_1)
      t_2)
     (if (<= t_0 0.0) t_3 (if (<= t_0 2e+264) (* t_2 t_1) t_3)))))

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 = (1.0 - ((h / l) * (pow(((M_m * D) / (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((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= -1e-167) {
		tmp = (fma((((-0.25 * (M_m * D)) / (l * d)) * (((0.5 / d) * D) * M_m)), h, 1.0) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 2e+264) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -1e-167)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(-0.25 * Float64(M_m * D)) / Float64(l * d)) * Float64(Float64(Float64(0.5 / d) * D) * M_m)), h, 1.0) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 2e+264)
		tmp = Float64(t_2 * t_1);
	else
		tmp = t_3;
	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[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-167], N[(N[(N[(N[(N[(N[(-0.25 * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$3, If[LessEqual[t$95$0, 2e+264], N[(t$95$2 * t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-167

   1. Initial program 88.7%

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

      1. lift-*.f64 N/A

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

      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 89.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 89.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 89.5

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

   6. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.6

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

   8. Applied rewrites 90.6%

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

   10. Applied rewrites 83.5%

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

   if -1e-167 < (*.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 2.00000000000000009e264 < (*.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 27.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 57.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell \cdot d} \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{M\_m \cdot M\_m}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 -1e+93)
     (* (* (/ (* M_m M_m) d) (* 0.125 (* D D))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 2e+264) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

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 = (1.0 - ((h / l) * (pow(((M_m * D) / (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 <= -1e+93) {
		tmp = (((M_m * M_m) / d) * (0.125 * (D * D))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+264) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((m_m * d_1) / (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 <= (-1d+93)) then
        tmp = (((m_m * m_m) / d) * (0.125d0 * (d_1 * d_1))) * sqrt((h / ((l * l) * l)))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+264) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_1
    end if
    code = tmp
end function

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 = (1.0 - ((h / l) * (Math.pow(((M_m * D) / (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 <= -1e+93) {
		tmp = (((M_m * M_m) / d) * (0.125 * (D * D))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+264) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_1;
	}
	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 = (1.0 - ((h / l) * (math.pow(((M_m * D) / (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 <= -1e+93:
		tmp = (((M_m * M_m) / d) * (0.125 * (D * D))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+264:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_1
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -1e+93)
		tmp = Float64(Float64(Float64(Float64(M_m * M_m) / d) * Float64(0.125 * Float64(D * D))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+264)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	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 = (1.0 - ((h / l) * ((((M_m * D) / (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 <= -1e+93)
		tmp = (((M_m * M_m) / d) * (0.125 * (D * D))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+264)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_1;
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e+93], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+264], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -1.00000000000000004e93

   1. Initial program 88.9%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.3%

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

   if -1.00000000000000004e93 < (*.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 2.00000000000000009e264 < (*.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 30.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.9

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

   5. Applied rewrites 25.9%

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

   7. Applied rewrites 54.6%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(\frac{D \cdot D}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(-0.125 \cdot \left(M\_m \cdot M\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 -1e+202)
     (* (* (/ (* D D) d) (sqrt (/ h (* (* l l) l)))) (* -0.125 (* M_m M_m)))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 2e+264) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

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 = (1.0 - ((h / l) * (pow(((M_m * D) / (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 <= -1e+202) {
		tmp = (((D * D) / d) * sqrt((h / ((l * l) * l)))) * (-0.125 * (M_m * M_m));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+264) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((m_m * d_1) / (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 <= (-1d+202)) then
        tmp = (((d_1 * d_1) / d) * sqrt((h / ((l * l) * l)))) * ((-0.125d0) * (m_m * m_m))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+264) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_1
    end if
    code = tmp
end function

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 = (1.0 - ((h / l) * (Math.pow(((M_m * D) / (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 <= -1e+202) {
		tmp = (((D * D) / d) * Math.sqrt((h / ((l * l) * l)))) * (-0.125 * (M_m * M_m));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+264) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_1;
	}
	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 = (1.0 - ((h / l) * (math.pow(((M_m * D) / (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 <= -1e+202:
		tmp = (((D * D) / d) * math.sqrt((h / ((l * l) * l)))) * (-0.125 * (M_m * M_m))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+264:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_1
	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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -1e+202)
		tmp = Float64(Float64(Float64(Float64(D * D) / d) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * Float64(-0.125 * Float64(M_m * M_m)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+264)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	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 = (1.0 - ((h / l) * ((((M_m * D) / (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 <= -1e+202)
		tmp = (((D * D) / d) * sqrt((h / ((l * l) * l)))) * (-0.125 * (M_m * M_m));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+264)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_1;
	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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e+202], N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+264], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -9.999999999999999e201

   1. Initial program 88.6%

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

   3. Taylor expanded in d around 0

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 29.5%

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

   if -9.999999999999999e201 < (*.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 2.00000000000000009e264 < (*.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 d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 52.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(\frac{D \cdot D}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{-1}{\ell \cdot h}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}} \cdot d\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (/ -1.0 (* l h)))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_1 -1e-167)
     (* (sqrt (sqrt (* t_0 t_0))) d)
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+264) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

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 = -1.0 / (l * h);
	double t_1 = (1.0 - ((h / l) * (pow(((M_m * D) / (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 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_1 <= -1e-167) {
		tmp = sqrt(sqrt((t_0 * t_0))) * d;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+264) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) / (l * h)
    t_1 = (1.0d0 - ((h / l) * ((((m_m * d_1) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_2 = abs((d / sqrt((l * h))))
    if (t_1 <= (-1d-167)) then
        tmp = sqrt(sqrt((t_0 * t_0))) * d
    else if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 2d+264) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_2
    end if
    code = tmp
end function

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 = -1.0 / (l * h);
	double t_1 = (1.0 - ((h / l) * (Math.pow(((M_m * D) / (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_2 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_1 <= -1e-167) {
		tmp = Math.sqrt(Math.sqrt((t_0 * t_0))) * d;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+264) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_2;
	}
	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 = -1.0 / (l * h)
	t_1 = (1.0 - ((h / l) * (math.pow(((M_m * D) / (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_2 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_1 <= -1e-167:
		tmp = math.sqrt(math.sqrt((t_0 * t_0))) * d
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2e+264:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_2
	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(-1.0 / Float64(l * h))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= -1e-167)
		tmp = Float64(sqrt(sqrt(Float64(t_0 * t_0))) * d);
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+264)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	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 = -1.0 / (l * h);
	t_1 = (1.0 - ((h / l) * ((((M_m * D) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_2 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_1 <= -1e-167)
		tmp = sqrt(sqrt((t_0 * t_0))) * d;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+264)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_2;
	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[(-1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-167], N[(N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+264], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -1e-167

   1. Initial program 88.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.4

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

   5. Applied rewrites 15.4%

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

   7. Applied rewrites 23.0%

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

   if -1e-167 < (*.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 2.00000000000000009e264 < (*.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 27.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 57.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\sqrt{\frac{-1}{\ell \cdot h} \cdot \frac{-1}{\ell \cdot h}}} \cdot d\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := M\_m \cdot \left(D \cdot 0.25\right)\\ t_3 := \left(M\_m \cdot D\right) \cdot 0.5\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m}{\frac{1}{h}} \cdot \frac{\frac{t\_2}{d}}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left|\frac{d}{t\_0}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{t\_3 \cdot t\_3}{d \cdot d}\right)}^{0.5} \cdot \frac{t\_2}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{t\_0}\\ \end{array} \end{array} \]

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 (* l h)))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 0.25)))
        (t_3 (* (* M_m D) 0.5)))
   (if (<= t_1 2e+264)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (/ (* (* (/ 0.5 d) D) M_m) (/ 1.0 h)) (/ (/ t_2 d) l))))
     (if (<= t_1 INFINITY)
       (fabs (/ d t_0))
       (/
        (*
         (fma (* (pow (/ (* t_3 t_3) (* d d)) 0.5) (/ t_2 (* (- l) d))) h 1.0)
         d)
        t_0)))))

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((l * h));
	double t_1 = (1.0 - ((h / l) * (pow(((M_m * D) / (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 * 0.25);
	double t_3 = (M_m * D) * 0.5;
	double tmp;
	if (t_1 <= 2e+264) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((0.5 / d) * D) * M_m) / (1.0 / h)) * ((t_2 / d) / l)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs((d / t_0));
	} else {
		tmp = (fma((pow(((t_3 * t_3) / (d * d)), 0.5) * (t_2 / (-l * d))), h, 1.0) * d) / t_0;
	}
	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(l * h))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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(M_m * Float64(D * 0.25))
	t_3 = Float64(Float64(M_m * D) * 0.5)
	tmp = 0.0
	if (t_1 <= 2e+264)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) / Float64(1.0 / h)) * Float64(Float64(t_2 / d) / l))));
	elseif (t_1 <= Inf)
		tmp = abs(Float64(d / t_0));
	else
		tmp = Float64(Float64(fma(Float64((Float64(Float64(t_3 * t_3) / Float64(d * d)) ^ 0.5) * Float64(t_2 / Float64(Float64(-l) * d))), h, 1.0) * d) / t_0);
	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[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M$95$m * D), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+264], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Abs[N[(d / t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Power[N[(N[(t$95$3 * t$95$3), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(t$95$2 / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * 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)))) < 2.00000000000000009e264

   1. Initial program 90.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. 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 90.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 90.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 90.2

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

   6. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 90.7

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

   10. Applied rewrites 90.7%

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

   if 2.00000000000000009e264 < (*.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 60.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   7. Applied rewrites 96.7%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      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 9.4%

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

   6. Applied rewrites 15.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 15.5

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 15.5

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow-prod-down N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. associate-/r* N/A

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

      20. div-inv N/A

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

      21. unpow-prod-down N/A

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

      22. associate-/l* N/A

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

      23. lower-pow.f64 N/A

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

   8. Applied rewrites 12.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\frac{1}{h}} \cdot \frac{\frac{M \cdot \left(D \cdot 0.25\right)}{d}}{\ell}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\left(\left(M \cdot D\right) \cdot 0.5\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{d \cdot d}\right)}^{0.5} \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{t\_0}{-d}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{t\_0}\right|\\ \end{array} \end{array} \]

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 (* l h)))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 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 0.0)
     (/ 1.0 (/ t_0 (- d)))
     (if (<= t_1 2e+264) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs (/ d t_0))))))

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((l * h));
	double t_1 = (1.0 - ((h / l) * (pow(((M_m * D) / (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 <= 0.0) {
		tmp = 1.0 / (t_0 / -d);
	} else if (t_1 <= 2e+264) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / t_0));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (1.0d0 - ((h / l) * ((((m_m * d_1) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= 0.0d0) then
        tmp = 1.0d0 / (t_0 / -d)
    else if (t_1 <= 2d+264) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / t_0))
    end if
    code = tmp
end function

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.sqrt((l * h));
	double t_1 = (1.0 - ((h / l) * (Math.pow(((M_m * D) / (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 <= 0.0) {
		tmp = 1.0 / (t_0 / -d);
	} else if (t_1 <= 2e+264) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / t_0));
	}
	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.sqrt((l * h))
	t_1 = (1.0 - ((h / l) * (math.pow(((M_m * D) / (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 <= 0.0:
		tmp = 1.0 / (t_0 / -d)
	elif t_1 <= 2e+264:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / t_0))
	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(l * h))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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 <= 0.0)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-d)));
	elseif (t_1 <= 2e+264)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / t_0));
	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 = sqrt((l * h));
	t_1 = (1.0 - ((h / l) * ((((M_m * D) / (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 <= 0.0)
		tmp = 1.0 / (t_0 / -d);
	elseif (t_1 <= 2e+264)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / t_0));
	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[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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, 0.0], N[(1.0 / N[(t$95$0 / (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+264], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / t$95$0), $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 83.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 20.6

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

   5. Applied rewrites 20.6%

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

   7. Applied rewrites 7.5%

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

   8. Taylor expanded in d around -inf

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

   10. Applied rewrites 21.7%

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

   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)))) < 2.00000000000000009e264

   1. Initial program 99.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 99.0%

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

   if 2.00000000000000009e264 < (*.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 24.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 21.6

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

   5. Applied rewrites 21.6%

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

   7. Applied rewrites 53.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot h}}{-d}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m}{\frac{1}{h}} \cdot \frac{\frac{M\_m \cdot \left(D \cdot 0.25\right)}{d}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

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
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      2e+264)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (-
     1.0
     (* (/ (* (* (/ 0.5 d) D) M_m) (/ 1.0 h)) (/ (/ (* M_m (* D 0.25)) d) l))))
   (fabs (/ d (sqrt (* l h))))))

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 tmp;
	if (((1.0 - ((h / l) * (pow(((M_m * D) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 2e+264) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((0.5 / d) * D) * M_m) / (1.0 / h)) * (((M_m * (D * 0.25)) / d) / l)));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (((1.0d0 - ((h / l) * ((((m_m * d_1) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= 2d+264) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((0.5d0 / d) * d_1) * m_m) / (1.0d0 / h)) * (((m_m * (d_1 * 0.25d0)) / d) / l)))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

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 tmp;
	if (((1.0 - ((h / l) * (Math.pow(((M_m * D) / (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+264) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((0.5 / d) * D) * M_m) / (1.0 / h)) * (((M_m * (D * 0.25)) / d) / l)));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	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):
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((M_m * D) / (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+264:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((0.5 / d) * D) * M_m) / (1.0 / h)) * (((M_m * (D * 0.25)) / d) / l)))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	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)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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+264)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) / Float64(1.0 / h)) * Float64(Float64(Float64(M_m * Float64(D * 0.25)) / d) / l))));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	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)
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((M_m * D) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= 2e+264)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((0.5 / d) * D) * M_m) / (1.0 / h)) * (((M_m * (D * 0.25)) / d) / l)));
	else
		tmp = abs((d / sqrt((l * h))));
	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_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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+264], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e264

   1. Initial program 90.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. 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 90.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 90.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 90.2

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

   6. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 90.7

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

   10. Applied rewrites 90.7%

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

   if 2.00000000000000009e264 < (*.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 24.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 21.6

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

   5. Applied rewrites 21.6%

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

   7. Applied rewrites 53.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\frac{1}{h}} \cdot \frac{\frac{M \cdot \left(D \cdot 0.25\right)}{d}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

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
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -1e+42)
   (* (sqrt (/ 1.0 (* l h))) d)
   (fabs (/ d (sqrt (* l h))))))

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 tmp;
	if (((1.0 - ((h / l) * (pow(((M_m * D) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e+42) {
		tmp = sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (((1.0d0 - ((h / l) * ((((m_m * d_1) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d+42)) then
        tmp = sqrt((1.0d0 / (l * h))) * d
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

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 tmp;
	if (((1.0 - ((h / l) * (Math.pow(((M_m * D) / (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+42) {
		tmp = Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	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):
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((M_m * D) / (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+42:
		tmp = math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	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)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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+42)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	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)
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((M_m * D) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e+42)
		tmp = sqrt((1.0 / (l * h))) * d;
	else
		tmp = abs((d / sqrt((l * h))));
	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_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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+42], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000004e42

   1. Initial program 89.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.8

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

   5. Applied rewrites 15.8%

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

   if -1.00000000000000004e42 < (*.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 64.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 63.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

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 (/ d (sqrt (* l h)))))
   (if (<=
        (*
         (- 1.0 (* (/ h l) (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        -1e+42)
     t_0
     (fabs t_0))))

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 = d / sqrt((l * h));
	double tmp;
	if (((1.0 - ((h / l) * (pow(((M_m * D) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e+42) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (((1.0d0 - ((h / l) * ((((m_m * d_1) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d+42)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

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 = d / Math.sqrt((l * h));
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((M_m * D) / (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+42) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	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 = d / math.sqrt((l * h))
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((M_m * D) / (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+42:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	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(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / 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+42)
		tmp = t_0;
	else
		tmp = abs(t_0);
	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 / sqrt((l * h));
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((M_m * D) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e+42)
		tmp = t_0;
	else
		tmp = abs(t_0);
	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[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $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+42], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000004e42

   1. Initial program 89.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.8

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

   5. Applied rewrites 15.8%

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

   7. Applied rewrites 11.3%

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

   if -1.00000000000000004e42 < (*.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 64.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 63.0%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := -0.5 \cdot \frac{h}{\ell}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(t\_2, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot D}{d} \cdot \frac{D}{d}, 1\right) \cdot t\_1\right) \cdot t\_3}{t\_0}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{t\_2}{d}, \frac{0.25}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot M\_m\right) \cdot M\_m\right), 1\right) \cdot t\_1\right) \cdot t\_3}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (* -0.5 (/ h l)))
        (t_3 (sqrt (- d))))
   (if (<= d -1.2e-65)
     (/
      (*
       (* (fma t_2 (* (/ (* (* (* M_m M_m) 0.25) D) d) (/ D d)) 1.0) t_1)
       t_3)
      t_0)
     (if (<= d -5e-310)
       (/
        (*
         (* (fma (/ t_2 d) (* (/ 0.25 d) (* (* (* D D) M_m) M_m)) 1.0) t_1)
         t_3)
        t_0)
       (/
        (*
         (fma
          (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
          h
          1.0)
         d)
        (* (sqrt h) (sqrt 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(-l);
	double t_1 = sqrt((d / h));
	double t_2 = -0.5 * (h / l);
	double t_3 = sqrt(-d);
	double tmp;
	if (d <= -1.2e-65) {
		tmp = ((fma(t_2, (((((M_m * M_m) * 0.25) * D) / d) * (D / d)), 1.0) * t_1) * t_3) / t_0;
	} else if (d <= -5e-310) {
		tmp = ((fma((t_2 / d), ((0.25 / d) * (((D * D) * M_m) * M_m)), 1.0) * t_1) * t_3) / t_0;
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(-l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(-0.5 * Float64(h / l))
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.2e-65)
		tmp = Float64(Float64(Float64(fma(t_2, Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * D) / d) * Float64(D / d)), 1.0) * t_1) * t_3) / t_0);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(Float64(fma(Float64(t_2 / d), Float64(Float64(0.25 / d) * Float64(Float64(Float64(D * D) * M_m) * M_m)), 1.0) * t_1) * t_3) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.2e-65], N[(N[(N[(N[(t$95$2 * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(N[(N[(t$95$2 / d), $MachinePrecision] * N[(N[(0.25 / d), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2000000000000001e-65

   1. Initial program 79.1%

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

   4. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 72.9

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

   6. Applied rewrites 72.9%

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

   if -1.2000000000000001e-65 < d < -4.999999999999985e-310

   1. Initial program 63.0%

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

   4. Applied rewrites 49.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 74.4%

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

   if -4.999999999999985e-310 < d

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      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 71.3%

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

   6. Applied rewrites 62.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, \frac{\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot D}{d} \cdot \frac{D}{d}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-0.5 \cdot \frac{h}{\ell}}{d}, \frac{0.25}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\\ \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot D}{d} \cdot \frac{D}{d}, 1\right) \cdot t\_0\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M\_m \cdot D\right)}{\ell \cdot d}, t\_1 \cdot h, 1\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (/ d h))) (t_1 (* (* (/ 0.5 d) D) M_m)))
   (if (<= l -4.3e+77)
     (/
      (*
       (*
        (fma (* -0.5 (/ h l)) (* (/ (* (* (* M_m M_m) 0.25) D) d) (/ D d)) 1.0)
        t_0)
       (sqrt (- d)))
      (sqrt (- l)))
     (if (<= l -5e-310)
       (*
        (* (fma (/ (* -0.25 (* M_m D)) (* l d)) (* t_1 h) 1.0) t_0)
        (sqrt (/ d l)))
       (/
        (* (fma (* t_1 (/ (* M_m (* D 0.25)) (* (- l) d))) h 1.0) d)
        (* (sqrt h) (sqrt 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((d / h));
	double t_1 = ((0.5 / d) * D) * M_m;
	double tmp;
	if (l <= -4.3e+77) {
		tmp = ((fma((-0.5 * (h / l)), (((((M_m * M_m) * 0.25) * D) / d) * (D / d)), 1.0) * t_0) * sqrt(-d)) / sqrt(-l);
	} else if (l <= -5e-310) {
		tmp = (fma(((-0.25 * (M_m * D)) / (l * d)), (t_1 * h), 1.0) * t_0) * sqrt((d / l));
	} else {
		tmp = (fma((t_1 * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(d / h))
	t_1 = Float64(Float64(Float64(0.5 / d) * D) * M_m)
	tmp = 0.0
	if (l <= -4.3e+77)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * D) / d) * Float64(D / d)), 1.0) * t_0) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(fma(Float64(Float64(-0.25 * Float64(M_m * D)) / Float64(l * d)), Float64(t_1 * h), 1.0) * t_0) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(t_1 * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[l, -4.3e+77], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(-0.25 * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * h), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.29999999999999991e77

   1. Initial program 65.2%

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

   4. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.6

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

   6. Applied rewrites 68.6%

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

   if -4.29999999999999991e77 < l < -4.999999999999985e-310

   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. 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 82.3%

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

   6. Applied rewrites 0.4%

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

   8. Applied rewrites 80.5%

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

   if -4.999999999999985e-310 < l

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      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 71.3%

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

   6. Applied rewrites 62.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, \frac{\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot D}{d} \cdot \frac{D}{d}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell \cdot d}, \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{t\_0}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-132}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{d \cdot d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\frac{-D}{\ell} \cdot \left(\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot D\right)\right) \cdot \sqrt{-d}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l))))
   (if (<= d -7e+98)
     (fabs (/ (/ d t_0) (sqrt (- h))))
     (if (<= d -1.85e-132)
       (*
        (*
         (fma
          (* -0.5 (/ h l))
          (/ (* (* D D) (* (* M_m M_m) 0.25)) (* d d))
          1.0)
         (sqrt (/ d l)))
        (sqrt (/ d h)))
       (if (<= d -1.35e-298)
         (/
          (*
           (*
            (/ (- D) l)
            (* (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (* M_m M_m)) D))
           (sqrt (- d)))
          t_0)
         (/
          (*
           (fma
            (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
            h
            1.0)
           d)
          (* (sqrt h) (sqrt 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(-l);
	double tmp;
	if (d <= -7e+98) {
		tmp = fabs(((d / t_0) / sqrt(-h)));
	} else if (d <= -1.85e-132) {
		tmp = (fma((-0.5 * (h / l)), (((D * D) * ((M_m * M_m) * 0.25)) / (d * d)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else if (d <= -1.35e-298) {
		tmp = (((-D / l) * (((-0.125 * sqrt((h / ((d * d) * d)))) * (M_m * M_m)) * D)) * sqrt(-d)) / t_0;
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(-l))
	tmp = 0.0
	if (d <= -7e+98)
		tmp = abs(Float64(Float64(d / t_0) / sqrt(Float64(-h))));
	elseif (d <= -1.85e-132)
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), Float64(Float64(Float64(D * D) * Float64(Float64(M_m * M_m) * 0.25)) / Float64(d * d)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (d <= -1.35e-298)
		tmp = Float64(Float64(Float64(Float64(Float64(-D) / l) * Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(M_m * M_m)) * D)) * sqrt(Float64(-d))) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(-l)], $MachinePrecision]}, If[LessEqual[d, -7e+98], N[Abs[N[(N[(d / t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1.85e-132], N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-298], N[(N[(N[(N[((-D) / l), $MachinePrecision] * N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7e98

   1. Initial program 71.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.9

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

   5. Applied rewrites 5.9%

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 79.2%

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

   if -7e98 < d < -1.8500000000000001e-132

   1. Initial program 85.1%

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

   4. Applied rewrites 68.0%

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

   if -1.8500000000000001e-132 < d < -1.3500000000000001e-298

   1. Initial program 59.2%

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 59.5%

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

   if -1.3500000000000001e-298 < d

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      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 70.9%

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

   6. Applied rewrites 61.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 79.7

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

   8. Applied rewrites 79.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-132}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{d \cdot d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\frac{-D}{\ell} \cdot \left(\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(M \cdot M\right)\right) \cdot D\right)\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{t\_0}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-132}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\frac{-D}{\ell} \cdot \left(\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot D\right)\right) \cdot \sqrt{-d}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l))))
   (if (<= d -7e+98)
     (fabs (/ (/ d t_0) (sqrt (- h))))
     (if (<= d -1.85e-132)
       (*
        (*
         (fma
          (/ (* (* (* (* D D) 0.25) (* M_m M_m)) h) (* (* d d) l))
          -0.5
          1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d -1.35e-298)
         (/
          (*
           (*
            (/ (- D) l)
            (* (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (* M_m M_m)) D))
           (sqrt (- d)))
          t_0)
         (/
          (*
           (fma
            (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
            h
            1.0)
           d)
          (* (sqrt h) (sqrt 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(-l);
	double tmp;
	if (d <= -7e+98) {
		tmp = fabs(((d / t_0) / sqrt(-h)));
	} else if (d <= -1.85e-132) {
		tmp = (fma((((((D * D) * 0.25) * (M_m * M_m)) * h) / ((d * d) * l)), -0.5, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -1.35e-298) {
		tmp = (((-D / l) * (((-0.125 * sqrt((h / ((d * d) * d)))) * (M_m * M_m)) * D)) * sqrt(-d)) / t_0;
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(-l))
	tmp = 0.0
	if (d <= -7e+98)
		tmp = abs(Float64(Float64(d / t_0) / sqrt(Float64(-h))));
	elseif (d <= -1.85e-132)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * 0.25) * Float64(M_m * M_m)) * h) / Float64(Float64(d * d) * l)), -0.5, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -1.35e-298)
		tmp = Float64(Float64(Float64(Float64(Float64(-D) / l) * Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(M_m * M_m)) * D)) * sqrt(Float64(-d))) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(-l)], $MachinePrecision]}, If[LessEqual[d, -7e+98], N[Abs[N[(N[(d / t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1.85e-132], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-298], N[(N[(N[(N[((-D) / l), $MachinePrecision] * N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7e98

   1. Initial program 71.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.9

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

   5. Applied rewrites 5.9%

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 79.2%

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

   if -7e98 < d < -1.8500000000000001e-132

   1. Initial program 85.1%

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

   4. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 65.7%

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

   if -1.8500000000000001e-132 < d < -1.3500000000000001e-298

   1. Initial program 59.2%

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 59.5%

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

   if -1.3500000000000001e-298 < d

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      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 70.9%

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

   6. Applied rewrites 61.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 79.7

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

   8. Applied rewrites 79.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-132}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\frac{-D}{\ell} \cdot \left(\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(M \cdot M\right)\right) \cdot D\right)\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{t\_0}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-123}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\left(\left(--0.125\right) \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \sqrt{-d}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l))))
   (if (<= d -7e+98)
     (fabs (/ (/ d t_0) (sqrt (- h))))
     (if (<= d -4.1e-123)
       (*
        (*
         (fma
          (/ (* (* (* (* D D) 0.25) (* M_m M_m)) h) (* (* d d) l))
          -0.5
          1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d -1.35e-298)
         (/
          (*
           (*
            (* (- -0.125) (sqrt (/ h (* (* d d) d))))
            (* (/ (* D D) l) (* M_m M_m)))
           (sqrt (- d)))
          t_0)
         (/
          (*
           (fma
            (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
            h
            1.0)
           d)
          (* (sqrt h) (sqrt 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(-l);
	double tmp;
	if (d <= -7e+98) {
		tmp = fabs(((d / t_0) / sqrt(-h)));
	} else if (d <= -4.1e-123) {
		tmp = (fma((((((D * D) * 0.25) * (M_m * M_m)) * h) / ((d * d) * l)), -0.5, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -1.35e-298) {
		tmp = (((-(-0.125) * sqrt((h / ((d * d) * d)))) * (((D * D) / l) * (M_m * M_m))) * sqrt(-d)) / t_0;
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(-l))
	tmp = 0.0
	if (d <= -7e+98)
		tmp = abs(Float64(Float64(d / t_0) / sqrt(Float64(-h))));
	elseif (d <= -4.1e-123)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * 0.25) * Float64(M_m * M_m)) * h) / Float64(Float64(d * d) * l)), -0.5, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -1.35e-298)
		tmp = Float64(Float64(Float64(Float64(Float64(-(-0.125)) * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(Float64(Float64(D * D) / l) * Float64(M_m * M_m))) * sqrt(Float64(-d))) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(-l)], $MachinePrecision]}, If[LessEqual[d, -7e+98], N[Abs[N[(N[(d / t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -4.1e-123], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-298], N[(N[(N[(N[((--0.125) * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7e98

   1. Initial program 71.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.9

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

   5. Applied rewrites 5.9%

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 79.2%

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

   if -7e98 < d < -4.1e-123

   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

   4. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 66.9%

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

   if -4.1e-123 < d < -1.3500000000000001e-298

   1. Initial program 58.0%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 58.2%

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

   if -1.3500000000000001e-298 < d

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      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 70.9%

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

   6. Applied rewrites 61.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 79.7

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

   8. Applied rewrites 79.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-123}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\left(\left(--0.125\right) \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{t\_0}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-123}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\left(\frac{M\_m \cdot M\_m}{\ell} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \sqrt{-d}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l))))
   (if (<= d -7e+98)
     (fabs (/ (/ d t_0) (sqrt (- h))))
     (if (<= d -4.1e-123)
       (*
        (*
         (fma
          (/ (* (* (* (* D D) 0.25) (* M_m M_m)) h) (* (* d d) l))
          -0.5
          1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d -1.35e-298)
         (/
          (*
           (*
            (* (/ (* M_m M_m) l) (* 0.125 (* D D)))
            (sqrt (/ h (* (* d d) d))))
           (sqrt (- d)))
          t_0)
         (/
          (*
           (fma
            (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
            h
            1.0)
           d)
          (* (sqrt h) (sqrt 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(-l);
	double tmp;
	if (d <= -7e+98) {
		tmp = fabs(((d / t_0) / sqrt(-h)));
	} else if (d <= -4.1e-123) {
		tmp = (fma((((((D * D) * 0.25) * (M_m * M_m)) * h) / ((d * d) * l)), -0.5, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -1.35e-298) {
		tmp = (((((M_m * M_m) / l) * (0.125 * (D * D))) * sqrt((h / ((d * d) * d)))) * sqrt(-d)) / t_0;
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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(-l))
	tmp = 0.0
	if (d <= -7e+98)
		tmp = abs(Float64(Float64(d / t_0) / sqrt(Float64(-h))));
	elseif (d <= -4.1e-123)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * 0.25) * Float64(M_m * M_m)) * h) / Float64(Float64(d * d) * l)), -0.5, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -1.35e-298)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) / l) * Float64(0.125 * Float64(D * D))) * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * sqrt(Float64(-d))) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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[(-l)], $MachinePrecision]}, If[LessEqual[d, -7e+98], N[Abs[N[(N[(d / t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -4.1e-123], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-298], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7e98

   1. Initial program 71.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.9

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

   5. Applied rewrites 5.9%

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 79.2%

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

   if -7e98 < d < -4.1e-123

   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

   4. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 66.9%

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

   if -4.1e-123 < d < -1.3500000000000001e-298

   1. Initial program 58.0%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in h around 0

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

   10. Applied rewrites 58.2%

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

   if -1.3500000000000001e-298 < d

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      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 70.9%

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

   6. Applied rewrites 61.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 79.7

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

   8. Applied rewrites 79.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-123}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(\left(\frac{M \cdot M}{\ell} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-138}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot D\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(\frac{M\_m}{d} \cdot M\_m\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= d -7e+98)
   (fabs (/ (/ d (sqrt (- l))) (sqrt (- h))))
   (if (<= d -3.2e-138)
     (*
      (*
       (fma (/ (* (* (* (* D D) 0.25) (* M_m M_m)) h) (* (* d d) l)) -0.5 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d -5e-310)
       (fma
        (* (* (* (- D) D) (sqrt (/ h (* (* l l) l)))) (* (/ M_m d) M_m))
        -0.125
        (* (sqrt (/ 1.0 (* l h))) (- d)))
       (/
        (*
         (fma
          (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
          h
          1.0)
         d)
        (* (sqrt h) (sqrt 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 tmp;
	if (d <= -7e+98) {
		tmp = fabs(((d / sqrt(-l)) / sqrt(-h)));
	} else if (d <= -3.2e-138) {
		tmp = (fma((((((D * D) * 0.25) * (M_m * M_m)) * h) / ((d * d) * l)), -0.5, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = fma((((-D * D) * sqrt((h / ((l * l) * l)))) * ((M_m / d) * M_m)), -0.125, (sqrt((1.0 / (l * h))) * -d));
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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)
	tmp = 0.0
	if (d <= -7e+98)
		tmp = abs(Float64(Float64(d / sqrt(Float64(-l))) / sqrt(Float64(-h))));
	elseif (d <= -3.2e-138)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * 0.25) * Float64(M_m * M_m)) * h) / Float64(Float64(d * d) * l)), -0.5, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = fma(Float64(Float64(Float64(Float64(-D) * D) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * Float64(Float64(M_m / d) * M_m)), -0.125, Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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_] := If[LessEqual[d, -7e+98], N[Abs[N[(N[(d / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -3.2e-138], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(N[((-D) * D), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7e98

   1. Initial program 71.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 5.9

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

   5. Applied rewrites 5.9%

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 79.2%

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

   if -7e98 < d < -3.2000000000000001e-138

   1. Initial program 84.3%

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

   4. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 64.6%

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

   if -3.2000000000000001e-138 < d < -4.999999999999985e-310

   1. Initial program 57.4%

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 50.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      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 71.3%

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

   6. Applied rewrites 62.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-138}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot D\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(\frac{M}{d} \cdot M\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-77}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot D\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(\frac{M\_m}{d} \cdot M\_m\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= d -2.05e-77)
   (fabs (/ d (* (sqrt (- h)) (sqrt (- l)))))
   (if (<= d -5e-310)
     (fma
      (* (* (* (- D) D) (sqrt (/ h (* (* l l) l)))) (* (/ M_m d) M_m))
      -0.125
      (* (sqrt (/ 1.0 (* l h))) (- d)))
     (/
      (*
       (fma
        (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
        h
        1.0)
       d)
      (* (sqrt h) (sqrt 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 tmp;
	if (d <= -2.05e-77) {
		tmp = fabs((d / (sqrt(-h) * sqrt(-l))));
	} else if (d <= -5e-310) {
		tmp = fma((((-D * D) * sqrt((h / ((l * l) * l)))) * ((M_m / d) * M_m)), -0.125, (sqrt((1.0 / (l * h))) * -d));
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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)
	tmp = 0.0
	if (d <= -2.05e-77)
		tmp = abs(Float64(d / Float64(sqrt(Float64(-h)) * sqrt(Float64(-l)))));
	elseif (d <= -5e-310)
		tmp = fma(Float64(Float64(Float64(Float64(-D) * D) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * Float64(Float64(M_m / d) * M_m)), -0.125, Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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_] := If[LessEqual[d, -2.05e-77], N[Abs[N[(d / N[(N[Sqrt[(-h)], $MachinePrecision] * N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(N[((-D) * D), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.04999999999999981e-77

   1. Initial program 78.8%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 55.9%

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

   9. Applied rewrites 67.6%

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

   if -2.04999999999999981e-77 < d < -4.999999999999985e-310

   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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 45.8%

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

   if -4.999999999999985e-310 < d

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      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 71.3%

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

   6. Applied rewrites 62.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-77}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-D\right) \cdot D\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot \left(\frac{M}{d} \cdot M\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{-1}{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}} \cdot d\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{-0.25 \cdot \left(M\_m \cdot D\right)}{\ell}}{d} \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (/ -1.0 (* l h))))
   (if (<= l -7.8e-169)
     (fabs (/ d (* (sqrt (- h)) (sqrt (- l)))))
     (if (<= l -4.1e-297)
       (* (sqrt (sqrt (* t_0 t_0))) d)
       (if (<= l 7.6e+69)
         (/
          (*
           (fma
            (* (/ (/ (* -0.25 (* M_m D)) l) d) (* (* (/ 0.5 d) D) M_m))
            h
            1.0)
           d)
          (sqrt (* l h)))
         (/ d (* (sqrt h) (sqrt 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 = -1.0 / (l * h);
	double tmp;
	if (l <= -7.8e-169) {
		tmp = fabs((d / (sqrt(-h) * sqrt(-l))));
	} else if (l <= -4.1e-297) {
		tmp = sqrt(sqrt((t_0 * t_0))) * d;
	} else if (l <= 7.6e+69) {
		tmp = (fma(((((-0.25 * (M_m * D)) / l) / d) * (((0.5 / d) * D) * M_m)), h, 1.0) * d) / sqrt((l * h));
	} else {
		tmp = d / (sqrt(h) * sqrt(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(-1.0 / Float64(l * h))
	tmp = 0.0
	if (l <= -7.8e-169)
		tmp = abs(Float64(d / Float64(sqrt(Float64(-h)) * sqrt(Float64(-l)))));
	elseif (l <= -4.1e-297)
		tmp = Float64(sqrt(sqrt(Float64(t_0 * t_0))) * d);
	elseif (l <= 7.6e+69)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(-0.25 * Float64(M_m * D)) / l) / d) * Float64(Float64(Float64(0.5 / d) * D) * M_m)), h, 1.0) * d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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[(-1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.8e-169], N[Abs[N[(d / N[(N[Sqrt[(-h)], $MachinePrecision] * N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -4.1e-297], N[(N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[LessEqual[l, 7.6e+69], N[(N[(N[(N[(N[(N[(N[(-0.25 * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.79999999999999954e-169

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 49.6%

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

   9. Applied rewrites 58.1%

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

   if -7.79999999999999954e-169 < l < -4.1000000000000002e-297

   1. Initial program 88.8%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 36.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.8%

      \[\leadsto \sqrt{\sqrt{\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

   if -4.1000000000000002e-297 < l < 7.60000000000000055e69

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 79.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\frac{1}{h}}}\right) \]

   6. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-\frac{\left(0.25 \cdot D\right) \cdot M}{\ell \cdot d}\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right)} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      3. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{4} \cdot D\right) \cdot M\right)}{\ell \cdot d}} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\frac{1}{4} \cdot D\right) \cdot M\right)}{\color{blue}{\ell \cdot d}} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{\mathsf{neg}\left(\left(\frac{1}{4} \cdot D\right) \cdot M\right)}{\ell}}{d}} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{\mathsf{neg}\left(\left(\frac{1}{4} \cdot D\right) \cdot M\right)}{\ell}}{d}} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{4} \cdot D\right) \cdot M\right)}{\ell}}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot D\right) \cdot M}\right)}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot M\right)}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \left(D \cdot M\right)}\right)}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{neg}\left(\frac{1}{4} \cdot \color{blue}{\left(M \cdot D\right)}\right)}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(M \cdot D\right)}}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(M \cdot D\right)}}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\color{blue}{\frac{-1}{4}} \cdot \left(M \cdot D\right)}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

      15. lower-*.f64 80.3

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-0.25 \cdot \color{blue}{\left(M \cdot D\right)}}{\ell}}{d} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

   8. Applied rewrites 80.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell}}{d}} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

   if 7.60000000000000055e69 < l

   1. Initial program 60.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 45.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\sqrt{\frac{-1}{\ell \cdot h} \cdot \frac{-1}{\ell \cdot h}}} \cdot d\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell}}{d} \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 68.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{M\_m \cdot M\_m}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= d -4e-121)
   (fabs (/ (/ d (sqrt (- l))) (sqrt (- h))))
   (if (<= d -5e-310)
     (* (* (/ (* M_m M_m) d) (* 0.125 (* D D))) (sqrt (/ h (* (* l l) l))))
     (/
      (*
       (fma
        (* (* (* (/ 0.5 d) D) M_m) (/ (* M_m (* D 0.25)) (* (- l) d)))
        h
        1.0)
       d)
      (* (sqrt h) (sqrt 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 tmp;
	if (d <= -4e-121) {
		tmp = fabs(((d / sqrt(-l)) / sqrt(-h)));
	} else if (d <= -5e-310) {
		tmp = (((M_m * M_m) / d) * (0.125 * (D * D))) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = (fma(((((0.5 / d) * D) * M_m) * ((M_m * (D * 0.25)) / (-l * d))), h, 1.0) * d) / (sqrt(h) * sqrt(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)
	tmp = 0.0
	if (d <= -4e-121)
		tmp = abs(Float64(Float64(d / sqrt(Float64(-l))) / sqrt(Float64(-h))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(M_m * M_m) / d) * Float64(0.125 * Float64(D * D))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * Float64(Float64(M_m * Float64(D * 0.25)) / Float64(Float64(-l) * d))), h, 1.0) * d) / Float64(sqrt(h) * sqrt(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_] := If[LessEqual[d, -4e-121], N[Abs[N[(N[(d / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.9999999999999999e-121

   1. Initial program 79.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 6.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 66.2%

      \[\leadsto \left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right| \]

   if -3.9999999999999999e-121 < d < -4.999999999999985e-310

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if -4.999999999999985e-310 < d

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      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 71.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\frac{1}{h}}}\right) \]

   6. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-\frac{\left(0.25 \cdot D\right) \cdot M}{\ell \cdot d}\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{4} \cdot D\right) \cdot M}{\ell \cdot d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

      8. lower-sqrt.f64 81.1

         \[\leadsto \frac{\mathsf{fma}\left(\left(-\frac{\left(0.25 \cdot D\right) \cdot M}{\ell \cdot d}\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   8. Applied rewrites 81.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-\frac{\left(0.25 \cdot D\right) \cdot M}{\ell \cdot d}\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{\frac{d}{\sqrt{-\ell}}}{\sqrt{-h}}\right|\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\left(-\ell\right) \cdot d}, h, 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 62.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{-1}{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}} \cdot d\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.25 \cdot \left(M\_m \cdot D\right)}{\ell \cdot d}, \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\_m\right) \cdot h, 1\right)}{\sqrt{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (/ -1.0 (* l h))))
   (if (<= l -7.8e-169)
     (fabs (/ d (* (sqrt (- h)) (sqrt (- l)))))
     (if (<= l -5e-310)
       (* (sqrt (sqrt (* t_0 t_0))) d)
       (if (<= l 7.6e+69)
         (*
          (/
           (fma
            (/ (* -0.25 (* M_m D)) (* l d))
            (* (* (* (/ 0.5 d) D) M_m) h)
            1.0)
           (sqrt (* l h)))
          d)
         (/ d (* (sqrt h) (sqrt 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 = -1.0 / (l * h);
	double tmp;
	if (l <= -7.8e-169) {
		tmp = fabs((d / (sqrt(-h) * sqrt(-l))));
	} else if (l <= -5e-310) {
		tmp = sqrt(sqrt((t_0 * t_0))) * d;
	} else if (l <= 7.6e+69) {
		tmp = (fma(((-0.25 * (M_m * D)) / (l * d)), ((((0.5 / d) * D) * M_m) * h), 1.0) / sqrt((l * h))) * d;
	} else {
		tmp = d / (sqrt(h) * sqrt(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(-1.0 / Float64(l * h))
	tmp = 0.0
	if (l <= -7.8e-169)
		tmp = abs(Float64(d / Float64(sqrt(Float64(-h)) * sqrt(Float64(-l)))));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(sqrt(Float64(t_0 * t_0))) * d);
	elseif (l <= 7.6e+69)
		tmp = Float64(Float64(fma(Float64(Float64(-0.25 * Float64(M_m * D)) / Float64(l * d)), Float64(Float64(Float64(Float64(0.5 / d) * D) * M_m) * h), 1.0) / sqrt(Float64(l * h))) * d);
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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[(-1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.8e-169], N[Abs[N[(d / N[(N[Sqrt[(-h)], $MachinePrecision] * N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[LessEqual[l, 7.6e+69], N[(N[(N[(N[(N[(-0.25 * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.79999999999999954e-169

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 6.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 6.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.6%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 58.1%

      \[\leadsto \left|\frac{-d}{\sqrt{-\ell} \cdot \left(-\sqrt{-h}\right)}\right| \]

   if -7.79999999999999954e-169 < l < -4.999999999999985e-310

   1. Initial program 86.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 33.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

      \[\leadsto \sqrt{\sqrt{\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

   if -4.999999999999985e-310 < l < 7.60000000000000055e69

   1. Initial program 82.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      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 80.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\frac{1}{h}}}\right) \]

   6. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-\frac{\left(0.25 \cdot D\right) \cdot M}{\ell \cdot d}\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), h, 1\right) \cdot d}{\sqrt{\ell \cdot h}}} \]

   8. Applied rewrites 79.6%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell \cdot d}, \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot h, 1\right)}{\sqrt{\ell \cdot h}}} \]

   if 7.60000000000000055e69 < l

   1. Initial program 60.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 45.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt{\frac{-1}{\ell \cdot h} \cdot \frac{-1}{\ell \cdot h}}} \cdot d\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell \cdot d}, \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot h, 1\right)}{\sqrt{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 49.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= l -5e-310)
   (fabs (/ d (* (sqrt (- h)) (sqrt (- l)))))
   (if (<= l 4.7e-228)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt h) (sqrt 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 tmp;
	if (l <= -5e-310) {
		tmp = fabs((d / (sqrt(-h) * sqrt(-l))));
	} else if (l <= 4.7e-228) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = abs((d / (sqrt(-h) * sqrt(-l))))
    else if (l <= 4.7d-228) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

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 tmp;
	if (l <= -5e-310) {
		tmp = Math.abs((d / (Math.sqrt(-h) * Math.sqrt(-l))));
	} else if (l <= 4.7e-228) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	tmp = 0
	if l <= -5e-310:
		tmp = math.fabs((d / (math.sqrt(-h) * math.sqrt(-l))))
	elif l <= 4.7e-228:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = abs(Float64(d / Float64(sqrt(Float64(-h)) * sqrt(Float64(-l)))));
	elseif (l <= 4.7e-228)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = abs((d / (sqrt(-h) * sqrt(-l))));
	elseif (l <= 4.7e-228)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(h) * sqrt(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_] := If[LessEqual[l, -5e-310], N[Abs[N[(d / N[(N[Sqrt[(-h)], $MachinePrecision] * N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.7e-228], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 12.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 51.6%

      \[\leadsto \left|\frac{-d}{\sqrt{-\ell} \cdot \left(-\sqrt{-h}\right)}\right| \]

   if -4.999999999999985e-310 < l < 4.7000000000000002e-228

   1. Initial program 83.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 58.7

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.7000000000000002e-228 < l

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 42.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{-h} \cdot \sqrt{-\ell}}\right|\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 45.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot h}}{-d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= l 4.7e-228)
   (/ 1.0 (/ (sqrt (* l h)) (- d)))
   (/ d (* (sqrt h) (sqrt 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 tmp;
	if (l <= 4.7e-228) {
		tmp = 1.0 / (sqrt((l * h)) / -d);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= 4.7d-228) then
        tmp = 1.0d0 / (sqrt((l * h)) / -d)
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

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 tmp;
	if (l <= 4.7e-228) {
		tmp = 1.0 / (Math.sqrt((l * h)) / -d);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	tmp = 0
	if l <= 4.7e-228:
		tmp = 1.0 / (math.sqrt((l * h)) / -d)
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	tmp = 0.0
	if (l <= 4.7e-228)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * h)) / Float64(-d)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	tmp = 0.0;
	if (l <= 4.7e-228)
		tmp = 1.0 / (sqrt((l * h)) / -d);
	else
		tmp = d / (sqrt(h) * sqrt(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_] := If[LessEqual[l, 4.7e-228], N[(1.0 / N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.7000000000000002e-228

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 12.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 12.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d \cdot \frac{d}{h}}}}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(\frac{1}{d} \cdot \sqrt{h \cdot \ell}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.3%

       \[\leadsto \frac{1}{\frac{\sqrt{\ell \cdot h}}{\color{blue}{-d}}} \]

   if 4.7000000000000002e-228 < l

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 42.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot h}}{-d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 45.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= l 4.7e-228)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt h) (sqrt 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 tmp;
	if (l <= 4.7e-228) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= 4.7d-228) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

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 tmp;
	if (l <= 4.7e-228) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	tmp = 0
	if l <= 4.7e-228:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	tmp = 0.0
	if (l <= 4.7e-228)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	tmp = 0.0;
	if (l <= 4.7e-228)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(h) * sqrt(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_] := If[LessEqual[l, 4.7e-228], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.7000000000000002e-228

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 44.9

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.7000000000000002e-228 < l

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 42.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

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 (/ d (sqrt (* l h))))

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) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d / sqrt((l * h))
end function

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) {
	return d / Math.sqrt((l * h));
}

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):
	return d / math.sqrt((l * h))

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)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
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_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 24.3

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 24.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 22.8%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))))