Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 80.0%

Time: 1.6min

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. d = -2.622893925870344e-96
  2. h = -3.585730383782278e+85
  3. l = 1.6400335792283288e+37
  4. M = 3.357478782450983e+121
  5. D = 4.568773676108101e-263

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: 67.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -6 \cdot 10^{-27}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(t\_0 \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -3.9 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, t\_0 \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) (t_1 (sqrt (- d))))
   (if (<= h -6e-27)
     (*
      (/ t_1 (sqrt (- l)))
      (* (sqrt (/ d h)) (+ 1.0 (* h (* t_0 (/ -0.5 l))))))
     (if (<= h -3.9e-306)
       (*
        (* (/ t_1 (sqrt (- h))) (sqrt (/ d l)))
        (- 1.0 (* 0.5 (* (pow (/ (* M_m (* D_m 0.5)) d) 2.0) (/ h l)))))
       (* d (/ (fma (/ h l) (* t_0 -0.5) 1.0) (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((D_m * (M_m * (0.5 / d))), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -6e-27) {
		tmp = (t_1 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + (h * (t_0 * (-0.5 / l)))));
	} else if (h <= -3.9e-306) {
		tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * (pow(((M_m * (D_m * 0.5)) / d), 2.0) * (h / l))));
	} else {
		tmp = d * (fma((h / l), (t_0 * -0.5), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -6e-27)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(t_0 * Float64(-0.5 / l))))));
	elseif (h <= -3.9e-306)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(t_0 * -0.5), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -6e-27], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -3.9e-306], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.0000000000000002e-27

   1. Initial program 57.5%

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

   3. Simplified 57.5%

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

      1. clear-num 57.5%

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

      2. associate-*l/ 57.5%

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

      3. *-un-lft-identity 57.5%

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

      4. frac-times 57.6%

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

      5. associate-/l* 57.5%

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

      6. *-un-lft-identity 57.5%

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

      7. times-frac 57.5%

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

      8. metadata-eval 57.5%

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

   6. Applied egg-rr 57.5%

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

      1. associate-/r/ 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-/l* 67.8%

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

      4. associate-*r* 67.8%

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

      5. associate-*r/ 67.8%

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

      6. *-commutative 67.8%

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

      7. associate-/l* 67.7%

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

      8. associate-/l* 67.7%

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

   8. Simplified 67.7%

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

      1. frac-2neg 67.7%

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

      2. sqrt-div 74.0%

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

   10. Applied egg-rr 74.0%

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

   if -6.0000000000000002e-27 < h < -3.9e-306

   1. Initial program 80.5%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 90.2%

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

      2. div-inv 90.2%

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

      3. associate-*l* 90.2%

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

      4. metadata-eval 90.2%

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

   8. Applied egg-rr 90.2%

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

   if -3.9e-306 < h

   1. Initial program 72.3%

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

   4. Applied egg-rr 73.3%

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

      1. distribute-rgt1-in 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*l/ 84.9%

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

      5. associate-/l* 84.8%

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

   6. Simplified 86.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -3.9 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(t\_1 \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, t\_1 \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (pow (* D_m (* M_m (/ 0.5 d))) 2.0)))
   (if (<= l -1.1e+42)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
     (if (<= l 1.35e-304)
       (* (* (sqrt (/ d h)) (+ 1.0 (* h (* t_1 (/ -0.5 l))))) t_0)
       (* d (/ (fma (/ h l) (* t_1 -0.5) 1.0) (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = pow((D_m * (M_m * (0.5 / d))), 2.0);
	double tmp;
	if (l <= -1.1e+42) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else if (l <= 1.35e-304) {
		tmp = (sqrt((d / h)) * (1.0 + (h * (t_1 * (-0.5 / l))))) * t_0;
	} else {
		tmp = d * (fma((h / l), (t_1 * -0.5), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0
	tmp = 0.0
	if (l <= -1.1e+42)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	elseif (l <= 1.35e-304)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(-0.5 / l))))) * t_0);
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(t_1 * -0.5), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1.1e+42], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e-304], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1000000000000001e42

   1. Initial program 60.7%

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

   3. Simplified 60.6%

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

      1. frac-2neg 60.6%

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

      2. sqrt-div 72.7%

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

   6. Applied egg-rr 72.7%

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

   if -1.1000000000000001e42 < l < 1.35000000000000005e-304

   1. Initial program 72.1%

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

   3. Simplified 68.4%

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

      1. clear-num 68.3%

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

      2. associate-*l/ 68.3%

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

      3. *-un-lft-identity 68.3%

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

      4. frac-times 72.1%

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

      5. associate-/l* 68.3%

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

      6. *-un-lft-identity 68.3%

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

      7. times-frac 68.3%

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

      8. metadata-eval 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-/r/ 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-/l* 76.6%

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

      4. associate-*r* 76.6%

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

      5. associate-*r/ 79.2%

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

      6. *-commutative 79.2%

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

      7. associate-/l* 79.1%

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

      8. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

   if 1.35000000000000005e-304 < l

   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

   4. Applied egg-rr 75.5%

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

      1. distribute-rgt1-in 83.6%

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

      2. +-commutative 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*l/ 86.7%

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

      5. associate-/l* 86.5%

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

   6. Simplified 88.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -3.9 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -3.9e-306)
   (*
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (* (pow (/ (* M_m (* D_m 0.5)) d) 2.0) (/ h l)))))
   (*
    d
    (/
     (fma (/ h l) (* (pow (* D_m (* M_m (/ 0.5 d))) 2.0) -0.5) 1.0)
     (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -3.9e-306) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * (pow(((M_m * (D_m * 0.5)) / d), 2.0) * (h / l))));
	} else {
		tmp = d * (fma((h / l), (pow((D_m * (M_m * (0.5 / d))), 2.0) * -0.5), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -3.9e-306)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64((Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0) * -0.5), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -3.9e-306], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -3.9e-306

   1. Initial program 69.1%

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

   3. Simplified 66.7%

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

      1. frac-2neg 66.7%

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

      2. sqrt-div 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. associate-*r/ 77.4%

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

      2. div-inv 77.4%

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

      3. associate-*l* 77.4%

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

      4. metadata-eval 77.4%

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

   8. Applied egg-rr 77.4%

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

   if -3.9e-306 < h

   1. Initial program 72.3%

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

   4. Applied egg-rr 73.3%

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

      1. distribute-rgt1-in 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*l/ 84.9%

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

      5. associate-/l* 84.8%

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

   6. Simplified 86.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.9 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(t\_0 \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, t\_0 \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* D_m (* M_m (/ 0.5 d))) 2.0)))
   (if (<= l 1.9e-304)
     (* (* (sqrt (/ d h)) (+ 1.0 (* h (* t_0 (/ -0.5 l))))) (sqrt (/ d l)))
     (* d (/ (fma (/ h l) (* t_0 -0.5) 1.0) (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((D_m * (M_m * (0.5 / d))), 2.0);
	double tmp;
	if (l <= 1.9e-304) {
		tmp = (sqrt((d / h)) * (1.0 + (h * (t_0 * (-0.5 / l))))) * sqrt((d / l));
	} else {
		tmp = d * (fma((h / l), (t_0 * -0.5), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0
	tmp = 0.0
	if (l <= 1.9e-304)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(t_0 * Float64(-0.5 / l))))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(t_0 * -0.5), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 1.9e-304], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8999999999999998e-304

   1. Initial program 67.7%

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

   3. Simplified 65.4%

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

      1. clear-num 65.3%

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

      2. associate-*l/ 65.3%

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

      3. *-un-lft-identity 65.3%

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

      4. frac-times 67.7%

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

      5. associate-/l* 65.3%

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

      6. *-un-lft-identity 65.3%

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

      7. times-frac 65.3%

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

      8. metadata-eval 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. associate-/r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-/l* 70.4%

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

      4. associate-*r* 70.4%

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

      5. associate-*r/ 72.0%

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

      6. *-commutative 72.0%

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

      7. associate-/l* 71.9%

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

      8. associate-/l* 71.9%

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

   8. Simplified 71.9%

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

   if 1.8999999999999998e-304 < l

   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

   4. Applied egg-rr 75.5%

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

      1. distribute-rgt1-in 83.6%

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

      2. +-commutative 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*l/ 86.7%

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

      5. associate-/l* 86.5%

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

   6. Simplified 88.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-86}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(h \cdot \ell + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5 \cdot {\left(D\_m \cdot \frac{M\_m \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -9e-86)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l -5e-310)
     (* d (exp (* (log1p (+ (* h l) -1.0)) (- 0.5))))
     (*
      (+ 1.0 (* h (/ (* -0.5 (pow (* D_m (/ (* M_m 0.5) d)) 2.0)) l)))
      (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9e-86) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= -5e-310) {
		tmp = d * exp((log1p(((h * l) + -1.0)) * -0.5));
	} else {
		tmp = (1.0 + (h * ((-0.5 * pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9e-86) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= -5e-310) {
		tmp = d * Math.exp((Math.log1p(((h * l) + -1.0)) * -0.5));
	} else {
		tmp = (1.0 + (h * ((-0.5 * Math.pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -9e-86:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= -5e-310:
		tmp = d * math.exp((math.log1p(((h * l) + -1.0)) * -0.5))
	else:
		tmp = (1.0 + (h * ((-0.5 * math.pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -9e-86)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= -5e-310)
		tmp = Float64(d * exp(Float64(log1p(Float64(Float64(h * l) + -1.0)) * Float64(-0.5))));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)) / l))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -9e-86], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Exp[N[(N[Log[1 + N[(N[(h * l), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * (-0.5)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.9999999999999995e-86

   1. Initial program 65.5%

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

   4. Applied egg-rr 27.6%

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

      1. associate-*l/ 27.6%

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

      2. associate-/l* 27.7%

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

      3. associate-*r* 27.7%

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

      4. associate-*r/ 27.8%

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

      5. *-commutative 27.8%

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

      6. associate-/l* 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in d around -inf 50.1%

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

      1. associate-*r* 50.1%

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

      2. neg-mul-1 50.1%

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

   9. Simplified 50.1%

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

   if -8.9999999999999995e-86 < l < -4.999999999999985e-310

   1. Initial program 73.5%

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

   3. Taylor expanded in d around inf 26.9%

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

      1. pow1/2 26.9%

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

      2. pow-to-exp 26.9%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 24.7%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 24.7%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 54.5%

         \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

      2. expm1-undefine 54.5%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

      3. add-exp-log 54.5%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   7. Applied egg-rr 54.5%

      \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   4. Applied egg-rr 73.8%

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

      1. distribute-rgt1-in 82.5%

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

      2. +-commutative 82.5%

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

      3. associate-*l/ 84.8%

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

      4. associate-/l* 84.1%

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

      5. associate-*r* 84.1%

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

      6. associate-*r/ 87.0%

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

      7. *-commutative 87.0%

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

      8. associate-/l* 85.6%

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

      9. associate-/r* 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-86}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(h \cdot \ell + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (*
     (sqrt (/ d h))
     (+ 1.0 (* h (* (pow (* D_m (* M_m (/ 0.5 d))) 2.0) (/ -0.5 l)))))
    (sqrt (/ d l)))
   (*
    (+ 1.0 (* h (/ (* -0.5 (pow (* D_m (/ (* M_m 0.5) d)) 2.0)) l)))
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (sqrt((d / h)) * (1.0 + (h * (pow((D_m * (M_m * (0.5 / d))), 2.0) * (-0.5 / l))))) * sqrt((d / l));
	} else {
		tmp = (1.0 + (h * ((-0.5 * pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (sqrt((d / h)) * (1.0d0 + (h * (((d_m * (m_m * (0.5d0 / d))) ** 2.0d0) * ((-0.5d0) / l))))) * sqrt((d / l))
    else
        tmp = (1.0d0 + (h * (((-0.5d0) * ((d_m * ((m_m * 0.5d0) / d)) ** 2.0d0)) / l))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (h * (Math.pow((D_m * (M_m * (0.5 / d))), 2.0) * (-0.5 / l))))) * Math.sqrt((d / l));
	} else {
		tmp = (1.0 + (h * ((-0.5 * Math.pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -5e-310:
		tmp = (math.sqrt((d / h)) * (1.0 + (h * (math.pow((D_m * (M_m * (0.5 / d))), 2.0) * (-0.5 / l))))) * math.sqrt((d / l))
	else:
		tmp = (1.0 + (h * ((-0.5 * math.pow((D_m * ((M_m * 0.5) / d)), 2.0)) / l))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64((Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0) * Float64(-0.5 / l))))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)) / l))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (sqrt((d / h)) * (1.0 + (h * (((D_m * (M_m * (0.5 / d))) ^ 2.0) * (-0.5 / l))))) * sqrt((d / l));
	else
		tmp = (1.0 + (h * ((-0.5 * ((D_m * ((M_m * 0.5) / d)) ^ 2.0)) / l))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 68.5%

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

   3. Simplified 66.2%

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

      1. clear-num 66.2%

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

      2. associate-*l/ 66.1%

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

      3. *-un-lft-identity 66.1%

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

      4. frac-times 68.5%

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

      5. associate-/l* 66.1%

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

      6. *-un-lft-identity 66.1%

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

      7. times-frac 66.1%

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

      8. metadata-eval 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. associate-/r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-/l* 70.4%

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

      4. associate-*r* 70.4%

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

      5. associate-*r/ 72.1%

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

      6. *-commutative 72.1%

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

      7. associate-/l* 72.0%

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

      8. associate-/l* 72.0%

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

   8. Simplified 72.0%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   4. Applied egg-rr 73.8%

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

      1. distribute-rgt1-in 82.5%

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

      2. +-commutative 82.5%

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

      3. associate-*l/ 84.8%

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

      4. associate-/l* 84.1%

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

      5. associate-*r* 84.1%

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

      6. associate-*r/ 87.0%

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

      7. *-commutative 87.0%

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

      8. associate-/l* 85.6%

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

      9. associate-/r* 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 78.2%

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

Alternative 7: 47.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-84}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(h \cdot \ell + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.7e-84)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l -5e-310)
     (* d (exp (* (log1p (+ (* h l) -1.0)) (- 0.5))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.7e-84) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= -5e-310) {
		tmp = d * exp((log1p(((h * l) + -1.0)) * -0.5));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.7e-84) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= -5e-310) {
		tmp = d * Math.exp((Math.log1p(((h * l) + -1.0)) * -0.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.7e-84:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= -5e-310:
		tmp = d * math.exp((math.log1p(((h * l) + -1.0)) * -0.5))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.7e-84)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= -5e-310)
		tmp = Float64(d * exp(Float64(log1p(Float64(Float64(h * l) + -1.0)) * Float64(-0.5))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.7e-84], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Exp[N[(N[Log[1 + N[(N[(h * l), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * (-0.5)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6999999999999999e-84

   1. Initial program 65.5%

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

   4. Applied egg-rr 27.6%

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

      1. associate-*l/ 27.6%

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

      2. associate-/l* 27.7%

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

      3. associate-*r* 27.7%

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

      4. associate-*r/ 27.8%

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

      5. *-commutative 27.8%

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

      6. associate-/l* 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in d around -inf 50.1%

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

      1. associate-*r* 50.1%

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

      2. neg-mul-1 50.1%

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

   9. Simplified 50.1%

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

   if -2.6999999999999999e-84 < l < -4.999999999999985e-310

   1. Initial program 73.5%

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

   3. Taylor expanded in d around inf 26.9%

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

      1. pow1/2 26.9%

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

      2. pow-to-exp 26.9%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 24.7%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 24.7%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 54.5%

         \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

      2. expm1-undefine 54.5%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

      3. add-exp-log 54.5%

         \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   7. Applied egg-rr 54.5%

      \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   3. Taylor expanded in d around inf 47.2%

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

      1. pow1/2 47.2%

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

      2. pow-to-exp 44.8%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 44.8%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. neg-log 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      2. *-commutative 44.8%

         \[\leadsto d \cdot e^{\log \left(\frac{1}{\color{blue}{\ell \cdot h}}\right) \cdot 0.5} \]

      3. associate-/l/ 44.8%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)} \cdot 0.5} \]

      4. pow-to-exp 47.2%

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

      5. pow1/2 47.2%

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

      6. *-un-lft-identity 47.2%

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

      7. sqrt-div 53.1%

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

      8. sqrt-div 53.1%

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

      9. metadata-eval 53.1%

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

      10. pow1/2 53.1%

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

      11. pow-flip 53.2%

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

      12. metadata-eval 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. *-lft-identity 53.2%

         \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   9. Simplified 53.2%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-84}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{\mathsf{log1p}\left(h \cdot \ell + -1\right) \cdot \left(-0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -3.8e-183)
   (* (- d) (pow (* h l) -0.5))
   (if (<= l -5e-310)
     (* d (pow (/ 1.0 (pow (* h l) 3.0)) 0.16666666666666666))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3.8e-183) {
		tmp = -d * pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * pow((1.0 / pow((h * l), 3.0)), 0.16666666666666666);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-3.8d-183)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else if (l <= (-5d-310)) then
        tmp = d * ((1.0d0 / ((h * l) ** 3.0d0)) ** 0.16666666666666666d0)
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3.8e-183) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * Math.pow((1.0 / Math.pow((h * l), 3.0)), 0.16666666666666666);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -3.8e-183:
		tmp = -d * math.pow((h * l), -0.5)
	elif l <= -5e-310:
		tmp = d * math.pow((1.0 / math.pow((h * l), 3.0)), 0.16666666666666666)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -3.8e-183)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(1.0 / (Float64(h * l) ^ 3.0)) ^ 0.16666666666666666));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -3.8e-183)
		tmp = -d * ((h * l) ^ -0.5);
	elseif (l <= -5e-310)
		tmp = d * ((1.0 / ((h * l) ^ 3.0)) ^ 0.16666666666666666);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -3.8e-183], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(1.0 / N[Power[N[(h * l), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7999999999999996e-183

   1. Initial program 66.6%

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

   4. Applied egg-rr 24.2%

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

      1. associate-*l/ 25.3%

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

      2. associate-/l* 25.4%

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

      3. associate-*r* 25.4%

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

      4. associate-*r/ 25.5%

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

      5. *-commutative 25.5%

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

      6. associate-/l* 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in d around -inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-in 46.5%

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

      3. unpow-1 46.5%

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

      4. metadata-eval 46.5%

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

      5. pow-sqr 46.5%

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

      6. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      7. rem-square-sqrt 46.3%

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

      8. fabs-sqr 46.3%

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

      9. rem-square-sqrt 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -3.7999999999999996e-183 < l < -4.999999999999985e-310

   1. Initial program 77.1%

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

   3. Taylor expanded in d around inf 34.4%

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

      1. pow1/2 34.4%

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

      2. pow-to-exp 34.4%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 30.0%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 30.0%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. neg-log 34.4%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      2. *-commutative 34.4%

         \[\leadsto d \cdot e^{\log \left(\frac{1}{\color{blue}{\ell \cdot h}}\right) \cdot 0.5} \]

      3. associate-/l/ 34.4%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)} \cdot 0.5} \]

      4. pow-to-exp 34.4%

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

      5. metadata-eval 34.4%

         \[\leadsto d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\left(3 \cdot 0.16666666666666666\right)}} \]

      6. metadata-eval 34.4%

         \[\leadsto d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(3 \cdot \color{blue}{\frac{0.3333333333333333}{2}}\right)} \]

      7. pow-pow 64.7%

         \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      8. associate-/l/ 64.7%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      9. *-commutative 64.7%

         \[\leadsto d \cdot {\left({\left(\frac{1}{\color{blue}{h \cdot \ell}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      10. cube-div 64.7%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{{1}^{3}}{{\left(h \cdot \ell\right)}^{3}}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      11. metadata-eval 64.7%

          \[\leadsto d \cdot {\left(\frac{\color{blue}{1}}{{\left(h \cdot \ell\right)}^{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      12. metadata-eval 64.7%

          \[\leadsto d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{\color{blue}{0.16666666666666666}} \]

   7. Applied egg-rr 64.7%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   3. Taylor expanded in d around inf 47.2%

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

      1. pow1/2 47.2%

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

      2. pow-to-exp 44.8%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 44.8%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. neg-log 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      2. *-commutative 44.8%

         \[\leadsto d \cdot e^{\log \left(\frac{1}{\color{blue}{\ell \cdot h}}\right) \cdot 0.5} \]

      3. associate-/l/ 44.8%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)} \cdot 0.5} \]

      4. pow-to-exp 47.2%

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

      5. pow1/2 47.2%

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

      6. *-un-lft-identity 47.2%

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

      7. sqrt-div 53.1%

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

      8. sqrt-div 53.1%

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

      9. metadata-eval 53.1%

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

      10. pow1/2 53.1%

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

      11. pow-flip 53.2%

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

      12. metadata-eval 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. *-lft-identity 53.2%

         \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   9. Simplified 53.2%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -8.5e-183)
   (* (- d) (pow (* h l) -0.5))
   (if (<= l -5e-310)
     (* d (cbrt (pow (/ (/ 1.0 h) l) 1.5)))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -8.5e-183) {
		tmp = -d * pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * cbrt(pow(((1.0 / h) / l), 1.5));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -8.5e-183) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * Math.cbrt(Math.pow(((1.0 / h) / l), 1.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -8.5e-183)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (l <= -5e-310)
		tmp = Float64(d * cbrt((Float64(Float64(1.0 / h) / l) ^ 1.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -8.5e-183], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[Power[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.49999999999999973e-183

   1. Initial program 66.6%

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

   4. Applied egg-rr 24.2%

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

      1. associate-*l/ 25.3%

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

      2. associate-/l* 25.4%

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

      3. associate-*r* 25.4%

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

      4. associate-*r/ 25.5%

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

      5. *-commutative 25.5%

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

      6. associate-/l* 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in d around -inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-in 46.5%

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

      3. unpow-1 46.5%

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

      4. metadata-eval 46.5%

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

      5. pow-sqr 46.5%

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

      6. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      7. rem-square-sqrt 46.3%

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

      8. fabs-sqr 46.3%

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

      9. rem-square-sqrt 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -8.49999999999999973e-183 < l < -4.999999999999985e-310

   1. Initial program 77.1%

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

   3. Taylor expanded in d around inf 34.4%

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

      1. add-cbrt-cube 47.4%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 47.4%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. pow1 47.4%

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

      4. pow1/2 47.4%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}} \]

      5. pow-prod-up 47.4%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}}} \]

      6. associate-/r* 47.4%

         \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}} \]

      7. metadata-eval 47.4%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}} \]

   5. Applied egg-rr 47.4%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   3. Taylor expanded in d around inf 47.2%

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

      1. pow1/2 47.2%

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

      2. pow-to-exp 44.8%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 44.8%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. neg-log 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      2. *-commutative 44.8%

         \[\leadsto d \cdot e^{\log \left(\frac{1}{\color{blue}{\ell \cdot h}}\right) \cdot 0.5} \]

      3. associate-/l/ 44.8%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)} \cdot 0.5} \]

      4. pow-to-exp 47.2%

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

      5. pow1/2 47.2%

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

      6. *-un-lft-identity 47.2%

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

      7. sqrt-div 53.1%

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

      8. sqrt-div 53.1%

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

      9. metadata-eval 53.1%

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

      10. pow1/2 53.1%

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

      11. pow-flip 53.2%

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

      12. metadata-eval 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. *-lft-identity 53.2%

         \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   9. Simplified 53.2%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-182}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4.2e-182)
   (* (- d) (pow (* h l) -0.5))
   (if (<= l -5e-310)
     (* d (sqrt (/ 1.0 (* h l))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.2e-182) {
		tmp = -d * pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4.2d-182)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else if (l <= (-5d-310)) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.2e-182) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (l <= -5e-310) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4.2e-182:
		tmp = -d * math.pow((h * l), -0.5)
	elif l <= -5e-310:
		tmp = d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4.2e-182)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (l <= -5e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4.2e-182)
		tmp = -d * ((h * l) ^ -0.5);
	elseif (l <= -5e-310)
		tmp = d * sqrt((1.0 / (h * l)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4.2e-182], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.2000000000000001e-182

   1. Initial program 66.6%

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

   4. Applied egg-rr 24.2%

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

      1. associate-*l/ 25.3%

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

      2. associate-/l* 25.4%

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

      3. associate-*r* 25.4%

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

      4. associate-*r/ 25.5%

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

      5. *-commutative 25.5%

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

      6. associate-/l* 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in d around -inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-in 46.5%

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

      3. unpow-1 46.5%

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

      4. metadata-eval 46.5%

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

      5. pow-sqr 46.5%

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

      6. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      7. rem-square-sqrt 46.3%

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

      8. fabs-sqr 46.3%

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

      9. rem-square-sqrt 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -4.2000000000000001e-182 < l < -4.999999999999985e-310

   1. Initial program 77.1%

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

   3. Taylor expanded in d around inf 34.4%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.9%

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

   3. Taylor expanded in d around inf 47.2%

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

      1. pow1/2 47.2%

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

      2. pow-to-exp 44.8%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right) \cdot 0.5}} \]

      3. log-rec 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   5. Applied egg-rr 44.8%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   6. Step-by-step derivation

      1. neg-log 44.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      2. *-commutative 44.8%

         \[\leadsto d \cdot e^{\log \left(\frac{1}{\color{blue}{\ell \cdot h}}\right) \cdot 0.5} \]

      3. associate-/l/ 44.8%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)} \cdot 0.5} \]

      4. pow-to-exp 47.2%

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

      5. pow1/2 47.2%

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

      6. *-un-lft-identity 47.2%

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

      7. sqrt-div 53.1%

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

      8. sqrt-div 53.1%

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

      9. metadata-eval 53.1%

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

      10. pow1/2 53.1%

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

      11. pow-flip 53.2%

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

      12. metadata-eval 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. *-lft-identity 53.2%

         \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   9. Simplified 53.2%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-182}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -1.8e-183) (* (- d) t_0) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -1.8e-183) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-1.8d-183)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -1.8e-183) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -1.8e-183:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -1.8e-183)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -1.8e-183)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -1.8e-183], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8000000000000001e-183

   1. Initial program 66.6%

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

   4. Applied egg-rr 24.2%

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

      1. associate-*l/ 25.3%

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

      2. associate-/l* 25.4%

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

      3. associate-*r* 25.4%

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

      4. associate-*r/ 25.5%

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

      5. *-commutative 25.5%

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

      6. associate-/l* 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in d around -inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-in 46.5%

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

      3. unpow-1 46.5%

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

      4. metadata-eval 46.5%

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

      5. pow-sqr 46.5%

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

      6. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      7. rem-square-sqrt 46.3%

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

      8. fabs-sqr 46.3%

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

      9. rem-square-sqrt 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.8000000000000001e-183 < l

   1. Initial program 73.5%

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

   4. Applied egg-rr 21.7%

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

      1. associate-*l/ 23.0%

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

      2. associate-/l* 23.0%

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

      3. associate-*r* 23.0%

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

      4. associate-*r/ 24.3%

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

      5. *-commutative 24.3%

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

      6. associate-/l* 23.7%

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

   6. Simplified 23.7%

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

   7. Taylor expanded in d around inf 45.5%

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

      1. unpow-1 45.5%

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

      2. metadata-eval 45.5%

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

      3. pow-sqr 45.4%

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

      4. rem-sqrt-square 44.8%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 44.8%

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

      6. fabs-sqr 44.8%

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

      7. rem-square-sqrt 44.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.3e-176)
   (* (- d) (pow (* h l) -0.5))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.3e-176) {
		tmp = -d * pow((h * l), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.3d-176)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.3e-176) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.3e-176:
		tmp = -d * math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.3e-176)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.3e-176)
		tmp = -d * ((h * l) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.3e-176], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.3000000000000001e-176

   1. Initial program 66.6%

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

   4. Applied egg-rr 24.2%

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

      1. associate-*l/ 25.3%

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

      2. associate-/l* 25.4%

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

      3. associate-*r* 25.4%

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

      4. associate-*r/ 25.5%

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

      5. *-commutative 25.5%

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

      6. associate-/l* 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in d around -inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-in 46.5%

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

      3. unpow-1 46.5%

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

      4. metadata-eval 46.5%

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

      5. pow-sqr 46.5%

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

      6. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      7. rem-square-sqrt 46.3%

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

      8. fabs-sqr 46.3%

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

      9. rem-square-sqrt 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -2.3000000000000001e-176 < l

   1. Initial program 73.5%

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

   3. Taylor expanded in d around inf 45.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (pow (* h l) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * pow((h * l), -0.5);
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * ((h * l) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.pow((h * l), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.pow((h * l), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

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

4. Applied egg-rr 22.6%

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

   1. associate-*l/ 23.9%

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

   2. associate-/l* 23.9%

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

   3. associate-*r* 23.9%

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

   4. associate-*r/ 24.7%

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

   5. *-commutative 24.7%

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

   6. associate-/l* 24.3%

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

6. Simplified 24.3%

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

7. Taylor expanded in d around inf 32.1%

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

   1. unpow-1 32.1%

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

   2. metadata-eval 32.1%

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

   3. pow-sqr 32.1%

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

   4. rem-sqrt-square 31.7%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 31.6%

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

   6. fabs-sqr 31.6%

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

   7. rem-square-sqrt 31.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 31.7%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 31.7%

    \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))