Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 79.4%

Time: 16.8s

Alternatives: 19

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -5.8e-304)
   (*
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
    (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
   (*
    (/ (fma (* h (pow (/ (* M_m (/ D d)) 2.0) 2.0)) (/ -0.5 l) 1.0) (sqrt h))
    (/ d (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 <= -5.8e-304) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (fma((h * pow(((M_m * (D / d)) / 2.0), 2.0)), (-0.5 / l), 1.0) / sqrt(h)) * (d / 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 <= -5.8e-304)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(fma(Float64(h * (Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0)), Float64(-0.5 / l), 1.0) / sqrt(h)) * Float64(d / 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, -5.8e-304], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.8e-304

   1. Initial program 73.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if -5.8e-304 < d

   1. Initial program 63.3%

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

   4. Applied rewrites 58.5%

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

   6. Applied rewrites 66.7%

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

   8. Applied rewrites 66.6%

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

   10. Applied rewrites 83.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M\_m \cdot M\_m}{\ell}\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M\_m \cdot M\_m\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -2e+40)
     (* (* (* (* -0.125 (/ (/ (* D D) d) d)) (* h (/ (* M_m M_m) l))) t_1) t_2)
     (if (<= t_0 INFINITY)
       (* t_1 t_2)
       (/
        (/
         (fma
          (* -0.25 (sqrt h))
          (* (* (* M_m M_m) D) D)
          (* (* l d) (sqrt (pow h -1.0))))
         l)
        (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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -2e+40) {
		tmp = (((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * t_1) * t_2;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 * t_2;
	} else {
		tmp = (fma((-0.25 * sqrt(h)), (((M_m * M_m) * D) * D), ((l * d) * sqrt(pow(h, -1.0)))) / l) / 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(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -2e+40)
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) / d) / d)) * Float64(h * Float64(Float64(M_m * M_m) / l))) * t_1) * t_2);
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(Float64(fma(Float64(-0.25 * sqrt(h)), Float64(Float64(Float64(M_m * M_m) * D) * D), Float64(Float64(l * d) * sqrt((h ^ -1.0)))) / l) / 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[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e+40], N[(N[(N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 * t$95$2), $MachinePrecision], N[(N[(N[(N[(-0.25 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] + N[(N[(l * d), $MachinePrecision] * N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[Sqrt[l], $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)))) < -2.00000000000000006e40

   1. Initial program 91.5%

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

   4. Applied rewrites 43.7%

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

   6. Applied rewrites 93.6%

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in d around 0

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 72.6

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

   11. Applied rewrites 72.6%

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

   if -2.00000000000000006e40 < (*.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 76.4%

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

   4. Applied rewrites 67.9%

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

   6. Applied rewrites 75.0%

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in d around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 73.6

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

   11. Applied rewrites 73.6%

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

   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-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 10.0%

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

   7. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 29.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M\_m \cdot M\_m}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

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 (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e+76)
   (*
    (* (* -0.125 (/ (/ (* D D) d) d)) (* h (/ (* M_m M_m) l)))
    (/ d (sqrt (* l h))))
   (* (sqrt (/ d h)) (sqrt (/ d 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 (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76) {
		tmp = ((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * (d / sqrt((l * h)));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / 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 (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d+76)) then
        tmp = (((-0.125d0) * (((d_1 * d_1) / d) / d)) * (h * ((m_m * m_m) / l))) * (d / sqrt((l * h)))
    else
        tmp = sqrt((d / h)) * sqrt((d / 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 (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76) {
		tmp = ((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / 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 ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76:
		tmp = ((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * (d / math.sqrt((l * h)))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / 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 (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e+76)
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) / d) / d)) * Float64(h * Float64(Float64(M_m * M_m) / l))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / 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 (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e+76)
		tmp = ((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * (d / sqrt((l * h)));
	else
		tmp = sqrt((d / h)) * sqrt((d / 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[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+76], N[(N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < -4.99999999999999991e76

   1. Initial program 91.4%

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

   4. Applied rewrites 44.3%

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

   6. Applied rewrites 93.6%

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

   8. Applied rewrites 49.0%

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

   9. Taylor expanded in d around 0

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 42.5

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

   11. Applied rewrites 42.5%

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

   if -4.99999999999999991e76 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

   4. Applied rewrites 55.2%

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

   6. Applied rewrites 60.4%

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in d around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 59.7

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

   11. Applied rewrites 59.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(M\_m \cdot M\_m\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

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 (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e+76)
   (* (* (* M_m M_m) (* (/ (* D D) d) -0.125)) (/ (sqrt (/ h l)) (fabs l)))
   (* (sqrt (/ d h)) (sqrt (/ d 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 (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76) {
		tmp = ((M_m * M_m) * (((D * D) / d) * -0.125)) * (sqrt((h / l)) / fabs(l));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / 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 (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d+76)) then
        tmp = ((m_m * m_m) * (((d_1 * d_1) / d) * (-0.125d0))) * (sqrt((h / l)) / abs(l))
    else
        tmp = sqrt((d / h)) * sqrt((d / 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 (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76) {
		tmp = ((M_m * M_m) * (((D * D) / d) * -0.125)) * (Math.sqrt((h / l)) / Math.abs(l));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / 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 ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+76:
		tmp = ((M_m * M_m) * (((D * D) / d) * -0.125)) * (math.sqrt((h / l)) / math.fabs(l))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / 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 (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e+76)
		tmp = Float64(Float64(Float64(M_m * M_m) * Float64(Float64(Float64(D * D) / d) * -0.125)) * Float64(sqrt(Float64(h / l)) / abs(l)));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / 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 (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e+76)
		tmp = ((M_m * M_m) * (((D * D) / d) * -0.125)) * (sqrt((h / l)) / abs(l));
	else
		tmp = sqrt((d / h)) * sqrt((d / 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[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+76], N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < -4.99999999999999991e76

   1. Initial program 91.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 39.9%

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

   9. Applied rewrites 43.7%

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

   if -4.99999999999999991e76 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

   4. Applied rewrites 55.2%

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

   6. Applied rewrites 60.4%

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in d around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 59.7

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

   11. Applied rewrites 59.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -3.75 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\frac{\frac{D}{d} \cdot M\_m}{2}\right)}^{2} \cdot 0.5}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= h -3.75e-180)
   (*
    (*
     (fma (/ (* (pow (/ (* (/ D d) M_m) 2.0) 2.0) 0.5) (- l)) h 1.0)
     (sqrt (/ d h)))
    (/ (sqrt (- d)) (sqrt (- l))))
   (if (<= h -2e-310)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (*
      (/ (fma (* h (pow (/ (* M_m (/ D d)) 2.0) 2.0)) (/ -0.5 l) 1.0) (sqrt h))
      (/ d (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 (h <= -3.75e-180) {
		tmp = (fma(((pow((((D / d) * M_m) / 2.0), 2.0) * 0.5) / -l), h, 1.0) * sqrt((d / h))) * (sqrt(-d) / sqrt(-l));
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (fma((h * pow(((M_m * (D / d)) / 2.0), 2.0)), (-0.5 / l), 1.0) / sqrt(h)) * (d / 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 (h <= -3.75e-180)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(Float64(Float64(D / d) * M_m) / 2.0) ^ 2.0) * 0.5) / Float64(-l)), h, 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(fma(Float64(h * (Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0)), Float64(-0.5 / l), 1.0) / sqrt(h)) * Float64(d / 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[h, -3.75e-180], N[(N[(N[(N[(N[(N[Power[N[(N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] / (-l)), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.75000000000000008e-180

   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 45.3%

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

   6. Applied rewrites 80.0%

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

   8. Applied rewrites 80.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 86.2

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

   10. Applied rewrites 86.2%

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

   if -3.75000000000000008e-180 < h < -1.999999999999994e-310

   1. Initial program 50.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 84.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.75 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}^{2} \cdot 0.5}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\frac{\frac{M\_m \cdot D}{d}}{2}\right)}^{2} \cdot 0.5}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= h -6.5e-180)
   (*
    (*
     (fma (/ (* (pow (/ (/ (* M_m D) d) 2.0) 2.0) 0.5) (- l)) h 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (if (<= h -2e-310)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (*
      (/ (fma (* h (pow (/ (* M_m (/ D d)) 2.0) 2.0)) (/ -0.5 l) 1.0) (sqrt h))
      (/ d (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 (h <= -6.5e-180) {
		tmp = (fma(((pow((((M_m * D) / d) / 2.0), 2.0) * 0.5) / -l), h, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (fma((h * pow(((M_m * (D / d)) / 2.0), 2.0)), (-0.5 / l), 1.0) / sqrt(h)) * (d / 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 (h <= -6.5e-180)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(Float64(Float64(M_m * D) / d) / 2.0) ^ 2.0) * 0.5) / Float64(-l)), h, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(fma(Float64(h * (Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0)), Float64(-0.5 / l), 1.0) / sqrt(h)) * Float64(d / 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[h, -6.5e-180], N[(N[(N[(N[(N[(N[Power[N[(N[(N[(M$95$m * D), $MachinePrecision] / d), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] / (-l)), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.50000000000000013e-180

   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 45.3%

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

   6. Applied rewrites 80.0%

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

   8. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 80.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.0

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

   10. Applied rewrites 80.0%

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

   if -6.50000000000000013e-180 < h < -1.999999999999994e-310

   1. Initial program 50.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 84.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\frac{\frac{M \cdot D}{d}}{2}\right)}^{2} \cdot 0.5}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot \frac{D}{d}}{2}\\ \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {t\_0}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 d)) 2.0)))
   (if (<= h -6.5e-180)
     (*
      (* (fma (* t_0 (* t_0 (/ -0.5 l))) h 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= h -2e-310)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (*
        (/ (fma (* h (pow t_0 2.0)) (/ -0.5 l) 1.0) (sqrt h))
        (/ d (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 / d)) / 2.0;
	double tmp;
	if (h <= -6.5e-180) {
		tmp = (fma((t_0 * (t_0 * (-0.5 / l))), h, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (fma((h * pow(t_0, 2.0)), (-0.5 / l), 1.0) / sqrt(h)) * (d / 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(Float64(M_m * Float64(D / d)) / 2.0)
	tmp = 0.0
	if (h <= -6.5e-180)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * Float64(-0.5 / l))), h, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(fma(Float64(h * (t_0 ^ 2.0)), Float64(-0.5 / l), 1.0) / sqrt(h)) * Float64(d / 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[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[h, -6.5e-180], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.50000000000000013e-180

   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 45.3%

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

   6. Applied rewrites 80.0%

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

   8. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 80.2

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

   10. Applied rewrites 80.2%

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

   if -6.50000000000000013e-180 < h < -1.999999999999994e-310

   1. Initial program 50.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 84.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot t\_0\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(D \cdot M\_m, \left(D \cdot M\_m\right) \cdot \left(\frac{h}{\ell \cdot d} \cdot -0.25\right), 1\right) \cdot t\_0\right) \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M\_m \cdot M\_m\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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))))
   (if (<= h -2.3e-142)
     (/ (* (* 1.0 t_0) (sqrt (- d))) (sqrt (- l)))
     (if (<= h -2e-310)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (if (<= h 2.7e+71)
         (/
          (*
           (* (fma (* D M_m) (* (* D M_m) (* (/ h (* l d)) -0.25)) 1.0) t_0)
           (sqrt d))
          (sqrt l))
         (/
          (/
           (fma
            (* -0.25 (sqrt h))
            (* (* (* M_m M_m) D) D)
            (* (* l d) (sqrt (pow h -1.0))))
           l)
          (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 tmp;
	if (h <= -2.3e-142) {
		tmp = ((1.0 * t_0) * sqrt(-d)) / sqrt(-l);
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (h <= 2.7e+71) {
		tmp = ((fma((D * M_m), ((D * M_m) * ((h / (l * d)) * -0.25)), 1.0) * t_0) * sqrt(d)) / sqrt(l);
	} else {
		tmp = (fma((-0.25 * sqrt(h)), (((M_m * M_m) * D) * D), ((l * d) * sqrt(pow(h, -1.0)))) / l) / 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))
	tmp = 0.0
	if (h <= -2.3e-142)
		tmp = Float64(Float64(Float64(1.0 * t_0) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (h <= 2.7e+71)
		tmp = Float64(Float64(Float64(fma(Float64(D * M_m), Float64(Float64(D * M_m) * Float64(Float64(h / Float64(l * d)) * -0.25)), 1.0) * t_0) * sqrt(d)) / sqrt(l));
	else
		tmp = Float64(Float64(fma(Float64(-0.25 * sqrt(h)), Float64(Float64(Float64(M_m * M_m) * D) * D), Float64(Float64(l * d) * sqrt((h ^ -1.0)))) / l) / 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]}, If[LessEqual[h, -2.3e-142], N[(N[(N[(1.0 * t$95$0), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.7e+71], N[(N[(N[(N[(N[(D * M$95$m), $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.25 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] + N[(N[(l * d), $MachinePrecision] * N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.30000000000000002e-142

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

   7. Applied rewrites 55.1%

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

   if -2.30000000000000002e-142 < h < -1.999999999999994e-310

   1. Initial program 58.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -1.999999999999994e-310 < h < 2.69999999999999997e71

   1. Initial program 75.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 75.8

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 69.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.0

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 71.3

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

   8. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

   if 2.69999999999999997e71 < h

   1. Initial program 47.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 47.1

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

   4. Applied rewrites 47.1%

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

   6. Applied rewrites 47.6%

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

   7. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 63.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(D \cdot M, \left(D \cdot M\right) \cdot \left(\frac{h}{\ell \cdot d} \cdot -0.25\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M\_m \cdot M\_m\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= h -2.3e-142)
   (/ (* (* 1.0 (sqrt (/ d h))) (sqrt (- d))) (sqrt (- l)))
   (if (<= h 5e-309)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (/
      (/
       (fma
        (* -0.25 (sqrt h))
        (* (* (* M_m M_m) D) D)
        (* (* l d) (sqrt (pow h -1.0))))
       l)
      (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 (h <= -2.3e-142) {
		tmp = ((1.0 * sqrt((d / h))) * sqrt(-d)) / sqrt(-l);
	} else if (h <= 5e-309) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (fma((-0.25 * sqrt(h)), (((M_m * M_m) * D) * D), ((l * d) * sqrt(pow(h, -1.0)))) / l) / 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 (h <= -2.3e-142)
		tmp = Float64(Float64(Float64(1.0 * sqrt(Float64(d / h))) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (h <= 5e-309)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.25 * sqrt(h)), Float64(Float64(Float64(M_m * M_m) * D) * D), Float64(Float64(l * d) * sqrt((h ^ -1.0)))) / l) / 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[h, -2.3e-142], N[(N[(N[(1.0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5e-309], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.25 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] + N[(N[(l * d), $MachinePrecision] * N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.30000000000000002e-142

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

   7. Applied rewrites 55.1%

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

   if -2.30000000000000002e-142 < h < 4.9999999999999995e-309

   1. Initial program 58.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if 4.9999999999999995e-309 < h

   1. Initial program 63.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 60.5%

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

   7. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 64.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.25 \cdot \sqrt{h}, \left(\left(M \cdot M\right) \cdot D\right) \cdot D, \left(\ell \cdot d\right) \cdot \sqrt{{h}^{-1}}\right)}{\ell}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-0.25 \cdot \sqrt{h}\right) \cdot \left(\frac{M\_m \cdot M\_m}{\ell} \cdot \left(D \cdot D\right)\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

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))))
   (if (<= d -7.5e-63)
     (/ (* (* 1.0 (sqrt (/ d l))) t_0) (sqrt (- h)))
     (if (<= d -5e-310)
       (/ (* (* 1.0 (sqrt (/ d h))) t_0) (sqrt (- l)))
       (if (<= d 1.15e-35)
         (/ (* (* -0.25 (sqrt h)) (* (/ (* M_m M_m) l) (* D D))) (sqrt l))
         (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))))

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);
	double tmp;
	if (d <= -7.5e-63) {
		tmp = ((1.0 * sqrt((d / l))) * t_0) / sqrt(-h);
	} else if (d <= -5e-310) {
		tmp = ((1.0 * sqrt((d / h))) * t_0) / sqrt(-l);
	} else if (d <= 1.15e-35) {
		tmp = ((-0.25 * sqrt(h)) * (((M_m * M_m) / l) * (D * D))) / sqrt(l);
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	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 = sqrt(-d)
    if (d <= (-7.5d-63)) then
        tmp = ((1.0d0 * sqrt((d / l))) * t_0) / sqrt(-h)
    else if (d <= (-5d-310)) then
        tmp = ((1.0d0 * sqrt((d / h))) * t_0) / sqrt(-l)
    else if (d <= 1.15d-35) then
        tmp = (((-0.25d0) * sqrt(h)) * (((m_m * m_m) / l) * (d_1 * d_1))) / sqrt(l)
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    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(-d);
	double tmp;
	if (d <= -7.5e-63) {
		tmp = ((1.0 * Math.sqrt((d / l))) * t_0) / Math.sqrt(-h);
	} else if (d <= -5e-310) {
		tmp = ((1.0 * Math.sqrt((d / h))) * t_0) / Math.sqrt(-l);
	} else if (d <= 1.15e-35) {
		tmp = ((-0.25 * Math.sqrt(h)) * (((M_m * M_m) / l) * (D * D))) / Math.sqrt(l);
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	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(-d)
	tmp = 0
	if d <= -7.5e-63:
		tmp = ((1.0 * math.sqrt((d / l))) * t_0) / math.sqrt(-h)
	elif d <= -5e-310:
		tmp = ((1.0 * math.sqrt((d / h))) * t_0) / math.sqrt(-l)
	elif d <= 1.15e-35:
		tmp = ((-0.25 * math.sqrt(h)) * (((M_m * M_m) / l) * (D * D))) / math.sqrt(l)
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	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))
	tmp = 0.0
	if (d <= -7.5e-63)
		tmp = Float64(Float64(Float64(1.0 * sqrt(Float64(d / l))) * t_0) / sqrt(Float64(-h)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(Float64(1.0 * sqrt(Float64(d / h))) * t_0) / sqrt(Float64(-l)));
	elseif (d <= 1.15e-35)
		tmp = Float64(Float64(Float64(-0.25 * sqrt(h)) * Float64(Float64(Float64(M_m * M_m) / l) * Float64(D * D))) / sqrt(l));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	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(-d);
	tmp = 0.0;
	if (d <= -7.5e-63)
		tmp = ((1.0 * sqrt((d / l))) * t_0) / sqrt(-h);
	elseif (d <= -5e-310)
		tmp = ((1.0 * sqrt((d / h))) * t_0) / sqrt(-l);
	elseif (d <= 1.15e-35)
		tmp = ((-0.25 * sqrt(h)) * (((M_m * M_m) / l) * (D * D))) / sqrt(l);
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	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[(-d)], $MachinePrecision]}, If[LessEqual[d, -7.5e-63], N[(N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(1.0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-35], N[(N[(N[(-0.25 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.5000000000000003e-63

   1. Initial program 79.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. sqrt-div N/A

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

   7. Applied rewrites 68.7%

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

   if -7.5000000000000003e-63 < d < -4.999999999999985e-310

   1. Initial program 62.3%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

   7. Applied rewrites 44.9%

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

   if -4.999999999999985e-310 < d < 1.1499999999999999e-35

   1. Initial program 56.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 56.1

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

   4. Applied rewrites 56.1%

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

   6. Applied rewrites 54.3%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 43.8

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

   9. Applied rewrites 43.8%

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

   if 1.1499999999999999e-35 < d

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 61.1

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 61.0%

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

   11. Applied rewrites 63.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-0.25 \cdot \sqrt{h}\right) \cdot \left(\frac{M \cdot M}{\ell} \cdot \left(D \cdot D\right)\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

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 (<= h -2.3e-142)
   (/ (* (* 1.0 (sqrt (/ d h))) (sqrt (- d))) (sqrt (- l)))
   (if (<= h -2e-310)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

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 (h <= -2.3e-142) {
		tmp = ((1.0 * sqrt((d / h))) * sqrt(-d)) / sqrt(-l);
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	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 (h <= (-2.3d-142)) then
        tmp = ((1.0d0 * sqrt((d / h))) * sqrt(-d)) / sqrt(-l)
    else if (h <= (-2d-310)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    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 (h <= -2.3e-142) {
		tmp = ((1.0 * Math.sqrt((d / h))) * Math.sqrt(-d)) / Math.sqrt(-l);
	} else if (h <= -2e-310) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	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 h <= -2.3e-142:
		tmp = ((1.0 * math.sqrt((d / h))) * math.sqrt(-d)) / math.sqrt(-l)
	elif h <= -2e-310:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	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 (h <= -2.3e-142)
		tmp = Float64(Float64(Float64(1.0 * sqrt(Float64(d / h))) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	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 (h <= -2.3e-142)
		tmp = ((1.0 * sqrt((d / h))) * sqrt(-d)) / sqrt(-l);
	elseif (h <= -2e-310)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	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[h, -2.3e-142], N[(N[(N[(1.0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.30000000000000002e-142

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

   7. Applied rewrites 55.1%

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

   if -2.30000000000000002e-142 < h < -1.999999999999994e-310

   1. Initial program 58.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 43.5

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

   7. Applied rewrites 43.5%

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

   9. Applied rewrites 43.5%

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

   11. Applied rewrites 46.8%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

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 (<= h -8.2e-144)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= h -2e-310)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

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 (h <= -8.2e-144) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	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 (h <= (-8.2d-144)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (h <= (-2d-310)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    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 (h <= -8.2e-144) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	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 h <= -8.2e-144:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif h <= -2e-310:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	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 (h <= -8.2e-144)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	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 (h <= -8.2e-144)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (h <= -2e-310)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	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[h, -8.2e-144], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.2e-144

   1. Initial program 80.4%

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

   4. Applied rewrites 46.1%

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

   6. Applied rewrites 81.4%

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in d around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 50.6

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

   11. Applied rewrites 50.6%

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

   if -8.2e-144 < h < -1.999999999999994e-310

   1. Initial program 57.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 57.0

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 72.8

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

   7. Applied rewrites 72.8%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 43.5

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

   7. Applied rewrites 43.5%

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

   9. Applied rewrites 43.5%

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

   11. Applied rewrites 46.8%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot \frac{D}{d}}{2}\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot \frac{-0.5}{\ell}\right), h, 1\right)\\ \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 d)) 2.0))
        (t_1 (fma (* t_0 (* t_0 (/ -0.5 l))) h 1.0)))
   (if (<= h -6.5e-180)
     (* (* t_1 (sqrt (/ d h))) (sqrt (/ d l)))
     (if (<= h -2e-310)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (* t_1 (/ 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 t_0 = (M_m * (D / d)) / 2.0;
	double t_1 = fma((t_0 * (t_0 * (-0.5 / l))), h, 1.0);
	double tmp;
	if (h <= -6.5e-180) {
		tmp = (t_1 * sqrt((d / h))) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = t_1 * (d / 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)
	t_0 = Float64(Float64(M_m * Float64(D / d)) / 2.0)
	t_1 = fma(Float64(t_0 * Float64(t_0 * Float64(-0.5 / l))), h, 1.0)
	tmp = 0.0
	if (h <= -6.5e-180)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	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[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(t$95$0 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]}, If[LessEqual[h, -6.5e-180], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.50000000000000013e-180

   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 45.3%

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

   6. Applied rewrites 80.0%

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

   8. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 80.2

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

   10. Applied rewrites 80.2%

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

   if -6.50000000000000013e-180 < h < -1.999999999999994e-310

   1. Initial program 50.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 63.8%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 80.9

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

   10. Applied rewrites 80.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

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 6.6e-164)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))

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 <= 6.6e-164) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	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 (d <= 6.6d-164) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    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 (d <= 6.6e-164) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	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 d <= 6.6e-164:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	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 <= 6.6e-164)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	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 (d <= 6.6e-164)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	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[d, 6.6e-164], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.6e-164

   1. Initial program 69.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if 6.6e-164 < d

   1. Initial program 67.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 48.9

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 48.9%

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

   11. Applied rewrites 53.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{M\_m \cdot \frac{D}{d}}{2}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.06 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(1 \cdot t\_3\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-202}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M\_m \cdot M\_m}{\ell}\right)\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(1 \cdot t\_2\right) \cdot t\_0}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(t\_1 \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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)))
        (t_1 (/ (* M_m (/ D d)) 2.0))
        (t_2 (sqrt (/ d h)))
        (t_3 (sqrt (/ d l))))
   (if (<= l -1.06e-137)
     (/ (* (* 1.0 t_3) t_0) (sqrt (- h)))
     (if (<= l -4.6e-202)
       (*
        (* (* (* -0.125 (/ (/ (* D D) d) d)) (* h (/ (* M_m M_m) l))) t_2)
        t_3)
       (if (<= l -1e-309)
         (/ (* (* 1.0 t_2) t_0) (sqrt (- l)))
         (* (fma (* t_1 (* t_1 (/ -0.5 l))) h 1.0) (/ 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 t_0 = sqrt(-d);
	double t_1 = (M_m * (D / d)) / 2.0;
	double t_2 = sqrt((d / h));
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -1.06e-137) {
		tmp = ((1.0 * t_3) * t_0) / sqrt(-h);
	} else if (l <= -4.6e-202) {
		tmp = (((-0.125 * (((D * D) / d) / d)) * (h * ((M_m * M_m) / l))) * t_2) * t_3;
	} else if (l <= -1e-309) {
		tmp = ((1.0 * t_2) * t_0) / sqrt(-l);
	} else {
		tmp = fma((t_1 * (t_1 * (-0.5 / l))), h, 1.0) * (d / 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)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(M_m * Float64(D / d)) / 2.0)
	t_2 = sqrt(Float64(d / h))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.06e-137)
		tmp = Float64(Float64(Float64(1.0 * t_3) * t_0) / sqrt(Float64(-h)));
	elseif (l <= -4.6e-202)
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) / d) / d)) * Float64(h * Float64(Float64(M_m * M_m) / l))) * t_2) * t_3);
	elseif (l <= -1e-309)
		tmp = Float64(Float64(Float64(1.0 * t_2) * t_0) / sqrt(Float64(-l)));
	else
		tmp = Float64(fma(Float64(t_1 * Float64(t_1 * Float64(-0.5 / l))), h, 1.0) * Float64(d / sqrt(Float64(l * h))));
	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[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.06e-137], N[(N[(N[(1.0 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.6e-202], N[(N[(N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(N[(1.0 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(t$95$1 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.06000000000000005e-137

   1. Initial program 69.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. sqrt-div N/A

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

   7. Applied rewrites 67.6%

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

   if -1.06000000000000005e-137 < l < -4.5999999999999997e-202

   1. Initial program 76.6%

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

   4. Applied rewrites 12.1%

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

   6. Applied rewrites 81.8%

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in d around 0

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 53.1

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

   11. Applied rewrites 53.1%

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

   if -4.5999999999999997e-202 < l < -1.000000000000002e-309

   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

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

   5. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

   7. Applied rewrites 50.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 63.8%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 80.9

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

   10. Applied rewrites 80.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right)}, h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.06 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-202}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{-259}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

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 (pow (* l h) -1.0))))
   (if (<= d -6.2e-259) (* (- d) t_0) (* t_0 d))))

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(pow((l * h), -1.0));
	double tmp;
	if (d <= -6.2e-259) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	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 = sqrt(((l * h) ** (-1.0d0)))
    if (d <= (-6.2d-259)) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    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(Math.pow((l * h), -1.0));
	double tmp;
	if (d <= -6.2e-259) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	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(math.pow((l * h), -1.0))
	tmp = 0
	if d <= -6.2e-259:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	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) ^ -1.0))
	tmp = 0.0
	if (d <= -6.2e-259)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	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) ^ -1.0));
	tmp = 0.0;
	if (d <= -6.2e-259)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	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[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.2e-259], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.1999999999999995e-259

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 73.2

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 52.3

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

   7. Applied rewrites 52.3%

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

   if -6.1999999999999995e-259 < d

   1. Initial program 64.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 44.3

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

   7. Applied rewrites 44.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{-259}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot \frac{D}{d}}{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(1 \cdot t\_1\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-280}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 d)) 2.0)) (t_1 (sqrt (/ d l))))
   (if (<= d -8e-25)
     (/ (* (* 1.0 t_1) (sqrt (- d))) (sqrt (- h)))
     (if (<= d -1.15e-280)
       (*
        (*
         (fma (/ (* (* 0.125 (* D D)) (/ (/ (* M_m M_m) d) d)) (- l)) h 1.0)
         (sqrt (/ d h)))
        t_1)
       (* (fma (* t_0 (* t_0 (/ -0.5 l))) h 1.0) (/ 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 t_0 = (M_m * (D / d)) / 2.0;
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -8e-25) {
		tmp = ((1.0 * t_1) * sqrt(-d)) / sqrt(-h);
	} else if (d <= -1.15e-280) {
		tmp = (fma((((0.125 * (D * D)) * (((M_m * M_m) / d) / d)) / -l), h, 1.0) * sqrt((d / h))) * t_1;
	} else {
		tmp = fma((t_0 * (t_0 * (-0.5 / l))), h, 1.0) * (d / 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)
	t_0 = Float64(Float64(M_m * Float64(D / d)) / 2.0)
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -8e-25)
		tmp = Float64(Float64(Float64(1.0 * t_1) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= -1.15e-280)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(Float64(M_m * M_m) / d) / d)) / Float64(-l)), h, 1.0) * sqrt(Float64(d / h))) * t_1);
	else
		tmp = Float64(fma(Float64(t_0 * Float64(t_0 * Float64(-0.5 / l))), h, 1.0) * Float64(d / sqrt(Float64(l * h))));
	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[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -8e-25], N[(N[(N[(1.0 * t$95$1), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.15e-280], N[(N[(N[(N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 * N[(t$95$0 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.00000000000000031e-25

   1. Initial program 77.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. sqrt-div N/A

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

   7. Applied rewrites 72.7%

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

   if -8.00000000000000031e-25 < d < -1.15e-280

   1. Initial program 67.2%

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

   4. Applied rewrites 31.6%

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

   6. Applied rewrites 67.2%

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

   8. Applied rewrites 67.3%

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

   9. Taylor expanded in d around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 50.0

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

   11. Applied rewrites 50.0%

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

   if -1.15e-280 < d

   1. Initial program 63.6%

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

   4. Applied rewrites 58.8%

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

   6. Applied rewrites 66.9%

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

   8. Applied rewrites 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \color{blue}{\frac{\frac{1}{2} \cdot -1}{\ell}}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      20. metadata-eval 80.4

          \[\leadsto \mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{\color{blue}{-0.5}}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 80.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right)}, h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-280}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}{-\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{2} \cdot \left(\frac{M \cdot \frac{D}{d}}{2} \cdot \frac{-0.5}{\ell}\right), h, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

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 (* (sqrt (pow (* l h) -1.0)) d))

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 sqrt(pow((l * h), -1.0)) * d;
}

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 = sqrt(((l * h) ** (-1.0d0))) * d
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 Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

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 math.sqrt(math.pow((l * h), -1.0)) * d

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(sqrt((Float64(l * h) ^ -1.0)) * d)
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 = sqrt(((l * h) ^ -1.0)) * d;
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[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lower-sqrt.f64 68.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Applied rewrites 68.5%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

5. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. 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 27.8

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

7. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

8. Final simplification 27.8%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
9. Add Preprocessing

Alternative 19: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

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 (* h l))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((h * l));
}

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((h * l))
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((h * l));
}

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((h * l))

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(h * l)))
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((h * l));
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lower-sqrt.f64 68.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Applied rewrites 68.5%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

5. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. 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 27.8

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

7. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
8. Step-by-step derivation

9. Applied rewrites 27.1%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
10. Step-by-step derivation

11. Applied rewrites 27.1%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024333 
(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)))))