Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 84.0%

Time: 37.8s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t\_0}{\sqrt{-\ell}}\\ t_2 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{-116}:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\frac{\ell}{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(t\_2 \cdot \frac{t\_2}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ t_0 (sqrt (- l)))) (t_2 (* D (/ M_m d))))
   (if (<= l -6e-116)
     (*
      t_1
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -5e-311)
       (*
        t_1
        (*
         (sqrt (/ d h))
         (+
          1.0
          (/ 1.0 (/ l (* (* h -0.5) (pow (/ (* D M_m) (* d 2.0)) 2.0)))))))
       (if (<= l 9.5e+139)
         (*
          (/ (sqrt d) (sqrt h))
          (*
           (sqrt (/ d l))
           (* (- h) (fma (* t_2 (/ t_2 l)) 0.125 (/ -1.0 h)))))
         (*
          d
          (/
           (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
           (* (sqrt h) (sqrt l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double t_1 = t_0 / sqrt(-l);
	double t_2 = D * (M_m / d);
	double tmp;
	if (l <= -6e-116) {
		tmp = t_1 * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -5e-311) {
		tmp = t_1 * (sqrt((d / h)) * (1.0 + (1.0 / (l / ((h * -0.5) * pow(((D * M_m) / (d * 2.0)), 2.0))))));
	} else if (l <= 9.5e+139) {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (-h * fma((t_2 * (t_2 / l)), 0.125, (-1.0 / h))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(t_0 / sqrt(Float64(-l)))
	t_2 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (l <= -6e-116)
		tmp = Float64(t_1 * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -5e-311)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(1.0 / Float64(l / Float64(Float64(h * -0.5) * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))));
	elseif (l <= 9.5e+139)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(Float64(-h) * fma(Float64(t_2 * Float64(t_2 / l)), 0.125, Float64(-1.0 / h)))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e-116], N[(t$95$1 * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(t$95$1 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(1.0 / N[(l / N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e+139], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[((-h) * N[(N[(t$95$2 * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.00000000000000053e-116

   1. Initial program 58.6%

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

   3. Simplified 59.6%

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

      1. frac-2neg 59.6%

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

      2. sqrt-div 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 82.8%

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

   8. Applied egg-rr 82.8%

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

   if -6.00000000000000053e-116 < l < -5.00000000000023e-311

   1. Initial program 74.0%

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

   3. Simplified 71.7%

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

      1. associate-*r/ 74.0%

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

      2. div-inv 74.0%

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

      3. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. metadata-eval 74.0%

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

      2. div-inv 74.0%

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

      3. associate-*r/ 71.7%

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

      4. associate-/l/ 71.7%

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

      5. associate-/l/ 71.7%

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

      6. associate-*l/ 82.9%

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

      7. associate-/l/ 82.9%

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

      8. *-commutative 82.9%

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

      9. clear-num 82.9%

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

   8. Applied egg-rr 85.2%

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

      1. frac-2neg 74.1%

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

      2. sqrt-div 76.3%

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

   10. Applied egg-rr 94.2%

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

   if -5.00000000000023e-311 < l < 9.5000000000000002e139

   1. Initial program 72.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in h around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. *-commutative 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

   7. Simplified 72.1%

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

      1. unpow2 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. times-frac 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. sqrt-div 83.9%

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

   11. Applied egg-rr 83.9%

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

   if 9.5000000000000002e139 < l

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   6. Applied egg-rr 73.1%

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

      1. unpow1 73.1%

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

      2. associate-*l/ 83.5%

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

      3. associate-/l* 83.3%

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

      4. +-commutative 83.3%

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

      5. associate-*r* 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*r/ 83.3%

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

      9. *-commutative 83.3%

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

      10. times-frac 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\frac{\ell}{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \frac{M\_m}{d}\\ t_2 := \left(-h\right) \cdot \mathsf{fma}\left(t\_1 \cdot \frac{t\_1}{\ell}, 0.125, \frac{-1}{h}\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot t\_2\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_3 \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_3 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (* D (/ M_m d)))
        (t_2 (* (- h) (fma (* t_1 (/ t_1 l)) 0.125 (/ -1.0 h))))
        (t_3 (sqrt (/ d l))))
   (if (<= l -5e+99)
     (* (sqrt (/ d h)) (* (/ t_0 (sqrt (- l))) t_2))
     (if (<= l -5e-311)
       (*
        t_3
        (*
         (/ t_0 (sqrt (- h)))
         (+ 1.0 (/ (* (* h -0.5) (pow (/ (* D M_m) (* d 2.0)) 2.0)) l))))
       (if (<= l 1.22e+140)
         (* (/ (sqrt d) (sqrt h)) (* t_3 t_2))
         (*
          d
          (/
           (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
           (* (sqrt h) (sqrt l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double t_1 = D * (M_m / d);
	double t_2 = -h * fma((t_1 * (t_1 / l)), 0.125, (-1.0 / h));
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -5e+99) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * t_2);
	} else if (l <= -5e-311) {
		tmp = t_3 * ((t_0 / sqrt(-h)) * (1.0 + (((h * -0.5) * pow(((D * M_m) / (d * 2.0)), 2.0)) / l)));
	} else if (l <= 1.22e+140) {
		tmp = (sqrt(d) / sqrt(h)) * (t_3 * t_2);
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(D * Float64(M_m / d))
	t_2 = Float64(Float64(-h) * fma(Float64(t_1 * Float64(t_1 / l)), 0.125, Float64(-1.0 / h)))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5e+99)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * t_2));
	elseif (l <= -5e-311)
		tmp = Float64(t_3 * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)) / l))));
	elseif (l <= 1.22e+140)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_3 * t_2));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-h) * N[(N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e+99], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(t$95$3 * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.22e+140], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.00000000000000008e99

   1. Initial program 47.6%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in h around -inf 35.7%

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

      1. mul-1-neg 35.7%

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

      2. *-commutative 35.7%

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

      3. distribute-rgt-neg-in 35.7%

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

   7. Simplified 51.8%

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

      1. unpow2 51.8%

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

      2. *-un-lft-identity 51.8%

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

      3. times-frac 58.5%

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

   9. Applied egg-rr 58.5%

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

       1. frac-2neg 60.2%

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

       2. sqrt-div 70.9%

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

   11. Applied egg-rr 77.8%

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

   if -5.00000000000000008e99 < l < -5.00000000000023e-311

   1. Initial program 71.0%

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

   3. Simplified 70.0%

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

      1. associate-*r/ 71.1%

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

      2. div-inv 71.1%

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

      3. metadata-eval 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. frac-2neg 70.0%

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

      2. sqrt-div 82.9%

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

   8. Applied egg-rr 84.0%

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

      1. associate-*l/ 89.1%

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

   10. Applied egg-rr 89.1%

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

   if -5.00000000000023e-311 < l < 1.22e140

   1. Initial program 72.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in h around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. *-commutative 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

   7. Simplified 72.1%

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

      1. unpow2 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. times-frac 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. sqrt-div 83.9%

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

   11. Applied egg-rr 83.9%

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

   if 1.22e140 < l

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   6. Applied egg-rr 73.1%

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

      1. unpow1 73.1%

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

      2. associate-*l/ 83.5%

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

      3. associate-/l* 83.3%

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

      4. +-commutative 83.3%

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

      5. associate-*r* 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*r/ 83.3%

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

      9. *-commutative 83.3%

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

      10. times-frac 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot t\_1\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -1.72 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_1 \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(t\_0 \cdot \frac{t\_0}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-299}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* D (/ M_m d)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (*
           (/ (sqrt (- d)) (sqrt (- h)))
           (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
          t_1)))
   (if (<= d -1.85e+121)
     t_2
     (if (<= d -1.72e-119)
       (*
        (sqrt (/ d h))
        (* t_1 (* (- h) (fma (* t_0 (/ t_0 l)) 0.125 (/ -1.0 h)))))
       (if (<= d -3.2e-299)
         t_2
         (*
          d
          (/
           (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
           (* (sqrt h) (sqrt l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = D * (M_m / d);
	double t_1 = sqrt((d / l));
	double t_2 = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_1;
	double tmp;
	if (d <= -1.85e+121) {
		tmp = t_2;
	} else if (d <= -1.72e-119) {
		tmp = sqrt((d / h)) * (t_1 * (-h * fma((t_0 * (t_0 / l)), 0.125, (-1.0 / h))));
	} else if (d <= -3.2e-299) {
		tmp = t_2;
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(D * Float64(M_m / d))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_1)
	tmp = 0.0
	if (d <= -1.85e+121)
		tmp = t_2;
	elseif (d <= -1.72e-119)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_1 * Float64(Float64(-h) * fma(Float64(t_0 * Float64(t_0 / l)), 0.125, Float64(-1.0 / h)))));
	elseif (d <= -3.2e-299)
		tmp = t_2;
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[d, -1.85e+121], t$95$2, If[LessEqual[d, -1.72e-119], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[((-h) * N[(N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-299], t$95$2, N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.85000000000000006e121 or -1.71999999999999993e-119 < d < -3.20000000000000008e-299

   1. Initial program 54.7%

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

   3. Simplified 54.7%

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

      1. frac-2neg 54.7%

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

      2. sqrt-div 74.3%

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

   6. Applied egg-rr 74.3%

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

   if -1.85000000000000006e121 < d < -1.71999999999999993e-119

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in h around -inf 65.7%

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

      1. mul-1-neg 65.7%

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

      2. *-commutative 65.7%

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

      3. distribute-rgt-neg-in 65.7%

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

   7. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. *-un-lft-identity 83.2%

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

      3. times-frac 88.9%

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

   9. Applied egg-rr 88.9%

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

   if -3.20000000000000008e-299 < d

   1. Initial program 65.3%

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

   3. Simplified 64.5%

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

   6. Applied egg-rr 78.0%

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

      1. unpow1 78.0%

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

      2. associate-*l/ 81.4%

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

      3. associate-/l* 80.6%

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

      4. +-commutative 80.6%

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

      5. associate-*r* 80.6%

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

      6. fma-define 80.6%

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

      7. *-commutative 80.6%

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

      8. associate-*r/ 80.6%

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

      9. *-commutative 80.6%

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

      10. times-frac 80.6%

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

   8. Simplified 80.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+121}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.72 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-299}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\sqrt{-d}}{\sqrt{-h}}\\ t_2 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;\left(t\_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(t\_2 \cdot \frac{t\_2}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-299}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \left(M\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ (sqrt (- d)) (sqrt (- h))))
        (t_2 (* D (/ M_m d))))
   (if (<= d -2.8e+123)
     (*
      (* t_1 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
      t_0)
     (if (<= d -1.4e-119)
       (*
        (sqrt (/ d h))
        (* t_0 (* (- h) (fma (* t_2 (/ t_2 l)) 0.125 (/ -1.0 h)))))
       (if (<= d -3.2e-299)
         (*
          t_0
          (*
           t_1
           (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D (* M_m 0.5)) d) 2.0))))))
         (*
          d
          (/
           (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
           (* (sqrt h) (sqrt l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt(-d) / sqrt(-h);
	double t_2 = D * (M_m / d);
	double tmp;
	if (d <= -2.8e+123) {
		tmp = (t_1 * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_0;
	} else if (d <= -1.4e-119) {
		tmp = sqrt((d / h)) * (t_0 * (-h * fma((t_2 * (t_2 / l)), 0.125, (-1.0 / h))));
	} else if (d <= -3.2e-299) {
		tmp = t_0 * (t_1 * (1.0 + ((h / l) * (-0.5 * pow(((D * (M_m * 0.5)) / d), 2.0)))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))
	t_2 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (d <= -2.8e+123)
		tmp = Float64(Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_0);
	elseif (d <= -1.4e-119)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(Float64(-h) * fma(Float64(t_2 * Float64(t_2 / l)), 0.125, Float64(-1.0 / h)))));
	elseif (d <= -3.2e-299)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * Float64(M_m * 0.5)) / d) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.8e+123], N[(N[(t$95$1 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -1.4e-119], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[((-h) * N[(N[(t$95$2 * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-299], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.80000000000000011e123

   1. Initial program 66.2%

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

   3. Simplified 68.6%

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

      1. frac-2neg 68.6%

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

      2. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if -2.80000000000000011e123 < d < -1.4e-119

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in h around -inf 65.7%

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

      1. mul-1-neg 65.7%

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

      2. *-commutative 65.7%

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

      3. distribute-rgt-neg-in 65.7%

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

   7. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. *-un-lft-identity 83.2%

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

      3. times-frac 88.9%

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

   9. Applied egg-rr 88.9%

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

   if -1.4e-119 < d < -3.20000000000000008e-299

   1. Initial program 43.6%

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

   3. Simplified 41.3%

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

      1. associate-*r/ 43.6%

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

      2. div-inv 43.6%

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

      3. metadata-eval 43.6%

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

   6. Applied egg-rr 43.6%

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

      1. frac-2neg 41.3%

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

      2. sqrt-div 63.8%

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

   8. Applied egg-rr 66.2%

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

   if -3.20000000000000008e-299 < d

   1. Initial program 65.3%

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

   3. Simplified 64.5%

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

   6. Applied egg-rr 78.0%

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

      1. unpow1 78.0%

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

      2. associate-*l/ 81.4%

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

      3. associate-/l* 80.6%

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

      4. +-commutative 80.6%

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

      5. associate-*r* 80.6%

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

      6. fma-define 80.6%

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

      7. *-commutative 80.6%

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

      8. associate-*r/ 80.6%

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

      9. *-commutative 80.6%

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

      10. times-frac 80.6%

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

   8. Simplified 80.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(t\_1 \cdot \frac{t\_1}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* D (/ M_m d))))
   (if (<= l -5e-311)
     (*
      t_0
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0))) l))))
     (if (<= l 7.1e+140)
       (*
        (/ (sqrt d) (sqrt h))
        (* t_0 (* (- h) (fma (* t_1 (/ t_1 l)) 0.125 (/ -1.0 h)))))
       (*
        d
        (/
         (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
         (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = D * (M_m / d);
	double tmp;
	if (l <= -5e-311) {
		tmp = t_0 * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h * (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0))) / l)));
	} else if (l <= 7.1e+140) {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (-h * fma((t_1 * (t_1 / l)), 0.125, (-1.0 / h))));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (l <= -5e-311)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0))) / l))));
	elseif (l <= 7.1e+140)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(Float64(-h) * fma(Float64(t_1 * Float64(t_1 / l)), 0.125, Float64(-1.0 / h)))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-311], N[(t$95$0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.1e+140], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[((-h) * N[(N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.00000000000023e-311

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

      1. associate-*l/ 67.4%

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

      2. *-commutative 67.4%

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

      3. add-sqr-sqrt 67.4%

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

      4. pow2 67.4%

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

      5. sqrt-pow1 67.4%

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

      6. metadata-eval 67.4%

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

      7. pow1 67.4%

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

      8. associate-/l/ 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. frac-2neg 63.2%

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

      2. sqrt-div 75.3%

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

   8. Applied egg-rr 79.5%

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

   if -5.00000000000023e-311 < l < 7.10000000000000027e140

   1. Initial program 72.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in h around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. *-commutative 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

   7. Simplified 72.1%

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

      1. unpow2 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. times-frac 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. sqrt-div 83.9%

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

   11. Applied egg-rr 83.9%

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

   if 7.10000000000000027e140 < l

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   6. Applied egg-rr 73.1%

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

      1. unpow1 73.1%

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

      2. associate-*l/ 83.5%

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

      3. associate-/l* 83.3%

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

      4. +-commutative 83.3%

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

      5. associate-*r* 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*r/ 83.3%

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

      9. *-commutative 83.3%

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

      10. times-frac 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \mathsf{fma}\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -5e-311)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0))) l))))
   (*
    d
    (/
     (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
     (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5e-311) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h * (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0))) / l)));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -5e-311)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -5e-311], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000023e-311

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

      1. associate-*l/ 67.4%

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

      2. *-commutative 67.4%

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

      3. add-sqr-sqrt 67.4%

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

      4. pow2 67.4%

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

      5. sqrt-pow1 67.4%

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

      6. metadata-eval 67.4%

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

      7. pow1 67.4%

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

      8. associate-/l/ 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. frac-2neg 63.2%

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

      2. sqrt-div 75.3%

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

   8. Applied egg-rr 79.5%

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

   if -5.00000000000023e-311 < l

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   6. Applied egg-rr 78.6%

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

      1. unpow1 78.6%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 81.2%

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

      4. +-commutative 81.2%

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

      5. associate-*r* 81.2%

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

      6. fma-define 81.2%

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

      7. *-commutative 81.2%

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

      8. associate-*r/ 81.2%

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

      9. *-commutative 81.2%

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

      10. times-frac 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+181}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2.05e+181)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l 6.8e-305)
     (*
      (/ 1.0 (sqrt (/ l d)))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0))) l))))
     (*
      d
      (/
       (fma (* (/ h l) -0.5) (pow (* (/ M_m 2.0) (/ D d)) 2.0) 1.0)
       (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.05e+181) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 6.8e-305) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0))) / l)));
	} else {
		tmp = d * (fma(((h / l) * -0.5), pow(((M_m / 2.0) * (D / d)), 2.0), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2.05e+181)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 6.8e-305)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2.05e+181], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e-305], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.05000000000000009e181

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -2.05000000000000009e181 < l < 6.8000000000000001e-305

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. add-sqr-sqrt 76.5%

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

      4. pow2 76.5%

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

      5. sqrt-pow1 76.5%

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

      6. metadata-eval 76.5%

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

      7. pow1 76.5%

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

      8. associate-/l/ 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. clear-num 76.5%

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

      2. sqrt-div 76.6%

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

      3. metadata-eval 76.6%

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

   8. Applied egg-rr 76.6%

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

   if 6.8000000000000001e-305 < l

   1. Initial program 66.4%

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

   3. Simplified 65.6%

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

   6. Applied egg-rr 79.3%

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

      1. unpow1 79.3%

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

      2. associate-*l/ 82.7%

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

      3. associate-/l* 81.9%

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

      4. +-commutative 81.9%

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

      5. associate-*r* 81.9%

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

      6. fma-define 81.9%

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

      7. *-commutative 81.9%

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

      8. associate-*r/ 81.9%

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

      9. *-commutative 81.9%

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

      10. times-frac 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+181}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\frac{\ell}{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2.4e+174)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l 1.5e-264)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ 1.0 (/ l (* (* h -0.5) (pow (/ (* D M_m) (* d 2.0)) 2.0))))))
      (sqrt (/ d l)))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.4e+174) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 1.5e-264) {
		tmp = (sqrt((d / h)) * (1.0 + (1.0 / (l / ((h * -0.5) * pow(((D * M_m) / (d * 2.0)), 2.0)))))) * sqrt((d / l));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.4d+174)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 1.5d-264) then
        tmp = (sqrt((d / h)) * (1.0d0 + (1.0d0 / (l / ((h * (-0.5d0)) * (((d_1 * m_m) / (d * 2.0d0)) ** 2.0d0)))))) * sqrt((d / l))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.4e+174) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 1.5e-264) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (1.0 / (l / ((h * -0.5) * Math.pow(((D * M_m) / (d * 2.0)), 2.0)))))) * Math.sqrt((d / l));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -2.4e+174:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 1.5e-264:
		tmp = (math.sqrt((d / h)) * (1.0 + (1.0 / (l / ((h * -0.5) * math.pow(((D * M_m) / (d * 2.0)), 2.0)))))) * math.sqrt((d / l))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2.4e+174)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 1.5e-264)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(1.0 / Float64(l / Float64(Float64(h * -0.5) * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -2.4e+174)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 1.5e-264)
		tmp = (sqrt((d / h)) * (1.0 + (1.0 / (l / ((h * -0.5) * (((D * M_m) / (d * 2.0)) ^ 2.0)))))) * sqrt((d / l));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2.4e+174], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-264], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(1.0 / N[(l / N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.3999999999999998e174

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -2.3999999999999998e174 < l < 1.5e-264

   1. Initial program 71.2%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 71.2%

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

      2. div-inv 71.2%

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

      3. metadata-eval 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. metadata-eval 71.2%

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

      2. div-inv 71.2%

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

      3. associate-*r/ 70.4%

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

      4. associate-/l/ 70.4%

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

      5. associate-/l/ 70.4%

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

      6. associate-*l/ 76.0%

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

      7. associate-/l/ 76.0%

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

      8. *-commutative 76.0%

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

      9. clear-num 76.0%

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

   8. Applied egg-rr 76.9%

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

   if 1.5e-264 < l

   1. Initial program 65.1%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 79.2%

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

      1. unpow1 79.2%

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

      2. associate-*r* 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*r/ 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*r/ 79.2%

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

      7. associate-*r* 79.2%

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

      8. associate-*r* 79.2%

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

      9. associate-/r* 79.2%

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

   8. Simplified 79.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-264}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\frac{\ell}{\left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -6.1 \cdot 10^{-221}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, {\left(\frac{M\_m \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.45e-111)
   (* d (- (pow (* l h) -0.5)))
   (if (<= d -6.1e-221)
     (* d (sqrt (log1p (expm1 (/ 1.0 (* l h))))))
     (if (<= d -4e-310)
       (* d (sqrt (log1p (expm1 (* l h)))))
       (*
        d
        (/
         (fma h (* (pow (/ (* M_m (/ D d)) 2.0) 2.0) (/ -0.5 l)) 1.0)
         (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.45e-111) {
		tmp = d * -pow((l * h), -0.5);
	} else if (d <= -6.1e-221) {
		tmp = d * sqrt(log1p(expm1((1.0 / (l * h)))));
	} else if (d <= -4e-310) {
		tmp = d * sqrt(log1p(expm1((l * h))));
	} else {
		tmp = d * (fma(h, (pow(((M_m * (D / d)) / 2.0), 2.0) * (-0.5 / l)), 1.0) / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.45e-111)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (d <= -6.1e-221)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(1.0 / Float64(l * h))))));
	elseif (d <= -4e-310)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(l * h)))));
	else
		tmp = Float64(d * Float64(fma(h, Float64((Float64(Float64(M_m * Float64(D / d)) / 2.0) ^ 2.0) * Float64(-0.5 / l)), 1.0) / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.45e-111], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -6.1e-221], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(l * h), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[Power[N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.45000000000000001e-111

   1. Initial program 73.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in d around inf 4.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 57.5%

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

      12. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

   if -1.45000000000000001e-111 < d < -6.10000000000000009e-221

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in d around inf 11.5%

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

      1. log1p-expm1-u 36.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

   7. Applied egg-rr 36.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

   if -6.10000000000000009e-221 < d < -3.999999999999988e-310

   1. Initial program 26.7%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in d around inf 7.3%

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

      1. add-exp-log 7.3%

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

      2. log-rec 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 7.6%

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

      3. sqr-neg 7.6%

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

      4. sqrt-unprod 6.8%

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

      5. add-sqr-sqrt 7.7%

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

      6. log1p-expm1-u 43.8%

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

      7. add-exp-log 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{h \cdot \ell}\right)\right)} \]

      8. *-commutative 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell \cdot h}\right)\right)} \]

   9. Applied egg-rr 43.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

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

   3. Simplified 64.2%

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

      1. associate-*l/ 66.0%

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

      2. *-commutative 66.0%

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

      3. add-sqr-sqrt 66.0%

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

      4. pow2 66.0%

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

      5. sqrt-pow1 66.0%

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

      6. metadata-eval 66.0%

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

      7. pow1 66.0%

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

      8. associate-/l/ 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. frac-2neg 0.0%

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

      2. sqrt-div 0.0%

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

   8. Applied egg-rr 0.0%

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

   10. Applied egg-rr 69.4%

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

       1. unpow1 69.4%

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

       2. associate-*l/ 70.2%

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

       3. associate-/l* 70.3%

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

       4. *-commutative 70.3%

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

       5. associate-/l* 70.3%

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

       6. *-commutative 70.3%

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

       7. associate-*l/ 70.3%

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

   12. Simplified 70.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -6.1 \cdot 10^{-221}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+181}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -5.5e+181)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -5e-309)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5.5e+181) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= -5e-309) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.5d+181)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= (-5d-309)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5.5e+181) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -5e-309) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -5.5e+181:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= -5e-309:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -5.5e+181)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -5e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -5.5e+181)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= -5e-309)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -5.5e+181], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.49999999999999991e181

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -5.49999999999999991e181 < l < -4.9999999999999995e-309

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   if -4.9999999999999995e-309 < l

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   6. Applied egg-rr 78.6%

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

      1. unpow1 78.6%

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

      2. associate-*r* 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*r/ 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*r/ 77.8%

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

      7. associate-*r* 77.8%

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

      8. associate-*r* 77.8%

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

      9. associate-/r* 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+181}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \left(M\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -6.5e+179)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l 2.2e-297)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D (* M_m 0.5)) d) 2.0))))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6.5e+179) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 2.2e-297) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow(((D * (M_m * 0.5)) / d), 2.0)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.5d+179)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 2.2d-297) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 * (m_m * 0.5d0)) / d) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6.5e+179) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 2.2e-297) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D * (M_m * 0.5)) / d), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -6.5e+179:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 2.2e-297:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D * (M_m * 0.5)) / d), 2.0)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -6.5e+179)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 2.2e-297)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * Float64(M_m * 0.5)) / d) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -6.5e+179)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 2.2e-297)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * (((D * (M_m * 0.5)) / d) ^ 2.0)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -6.5e+179], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e-297], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.50000000000000052e179

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -6.50000000000000052e179 < l < 2.1999999999999998e-297

   1. Initial program 70.7%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 70.7%

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

      2. div-inv 70.7%

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

      3. metadata-eval 70.7%

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

   6. Applied egg-rr 70.7%

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

   if 2.1999999999999998e-297 < l

   1. Initial program 66.1%

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

   3. Simplified 65.3%

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

   6. Applied egg-rr 79.1%

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

      1. unpow1 79.1%

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

      2. associate-*r* 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-*r/ 79.9%

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

      5. *-commutative 79.9%

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

      6. associate-*r/ 79.1%

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

      7. associate-*r* 79.1%

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

      8. associate-*r* 79.1%

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

      9. associate-/r* 79.1%

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

   8. Simplified 79.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+177}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.25e+177)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l 1.7e-263)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* h (* -0.5 (pow (* D (* M_m (/ 0.5 d))) 2.0))) l))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.25e+177) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 1.7e-263) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * pow((D * (M_m * (0.5 / d))), 2.0))) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.25d+177)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 1.7d-263) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.5d0) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.25e+177) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 1.7e-263) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * Math.pow((D * (M_m * (0.5 / d))), 2.0))) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1.25e+177:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 1.7e-263:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * math.pow((D * (M_m * (0.5 / d))), 2.0))) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.25e+177)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 1.7e-263)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0))) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1.25e+177)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 1.7e-263)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * ((D * (M_m * (0.5 / d))) ^ 2.0))) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.25e+177], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e-263], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.2500000000000001e177

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -1.2500000000000001e177 < l < 1.70000000000000002e-263

   1. Initial program 71.2%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 76.0%

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

      2. *-commutative 76.0%

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

      3. add-sqr-sqrt 76.0%

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

      4. pow2 76.0%

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

      5. sqrt-pow1 76.0%

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

      6. metadata-eval 76.0%

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

      7. pow1 76.0%

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

      8. associate-/l/ 76.0%

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

   6. Applied egg-rr 76.0%

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

   7. Taylor expanded in D around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. associate-*l/ 76.9%

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

      3. *-commutative 76.9%

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

      4. associate-*l* 76.0%

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

   9. Simplified 76.0%

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

   if 1.70000000000000002e-263 < l

   1. Initial program 65.1%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 79.2%

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

      1. unpow1 79.2%

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

      2. associate-*r* 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*r/ 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*r/ 79.2%

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

      7. associate-*r* 79.2%

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

      8. associate-*r* 79.2%

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

      9. associate-/r* 79.2%

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

   8. Simplified 79.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+177}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+182}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2.6e+182)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l 1.5e-264)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0))) l))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.6e+182) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 1.5e-264) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0))) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.6d+182)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 1.5d-264) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.5d0) * ((d_1 * (m_m / (d * 2.0d0))) ** 2.0d0))) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.6e+182) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 1.5e-264) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * Math.pow((D * (M_m / (d * 2.0))), 2.0))) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -2.6e+182:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 1.5e-264:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * math.pow((D * (M_m / (d * 2.0))), 2.0))) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2.6e+182)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 1.5e-264)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0))) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -2.6e+182)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 1.5e-264)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * ((D * (M_m / (d * 2.0))) ^ 2.0))) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2.6e+182], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-264], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6e182

   1. Initial program 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 58.3%

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

      6. neg-mul-1 58.3%

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

   7. Simplified 58.3%

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

   if -2.6e182 < l < 1.5e-264

   1. Initial program 71.2%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 76.0%

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

      2. *-commutative 76.0%

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

      3. add-sqr-sqrt 76.0%

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

      4. pow2 76.0%

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

      5. sqrt-pow1 76.0%

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

      6. metadata-eval 76.0%

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

      7. pow1 76.0%

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

      8. associate-/l/ 76.0%

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

   6. Applied egg-rr 76.0%

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

   if 1.5e-264 < l

   1. Initial program 65.1%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 79.2%

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

      1. unpow1 79.2%

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

      2. associate-*r* 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*r/ 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*r/ 79.2%

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

      7. associate-*r* 79.2%

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

      8. associate-*r* 79.2%

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

      9. associate-/r* 79.2%

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

   8. Simplified 79.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+182}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-27}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -8e-27)
   (* d (- (pow (* l h) -0.5)))
   (if (<= l -5e-311)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (* -0.125 (* (/ h l) (pow (* D (/ M_m d)) 2.0)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -8e-27) {
		tmp = d * -pow((l * h), -0.5);
	} else if (l <= -5e-311) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (-0.125 * ((h / l) * pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-8d-27)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else if (l <= (-5d-311)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * ((-0.125d0) * ((h / l) * ((d_1 * (m_m / d)) ** 2.0d0))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_1 * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -8e-27) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (l <= -5e-311) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (-0.125 * ((h / l) * Math.pow((D * (M_m / d)), 2.0))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * Math.pow((D * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -8e-27:
		tmp = d * -math.pow((l * h), -0.5)
	elif l <= -5e-311:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (-0.125 * ((h / l) * math.pow((D * (M_m / d)), 2.0))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (((h / l) * -0.5) * math.pow((D * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -8e-27)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (l <= -5e-311)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(-0.125 * Float64(Float64(h / l) * (Float64(D * Float64(M_m / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -8e-27)
		tmp = d * -((l * h) ^ -0.5);
	elseif (l <= -5e-311)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (-0.125 * ((h / l) * ((D * (M_m / d)) ^ 2.0))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((h / l) * -0.5) * ((D * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -8e-27], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-311], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.0000000000000003e-27

   1. Initial program 55.0%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in d around inf 4.1%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 55.7%

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

      12. neg-mul-1 55.7%

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

   8. Simplified 55.7%

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

   if -8.0000000000000003e-27 < l < -5.00000000000023e-311

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in M around inf 31.4%

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

      1. associate-*r* 33.0%

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

      2. times-frac 32.9%

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

      3. *-commutative 32.9%

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

      4. associate-/l* 31.4%

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

      5. unpow2 31.4%

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

      6. unpow2 31.4%

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

      7. unpow2 31.4%

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

      8. times-frac 41.7%

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

      9. swap-sqr 46.7%

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

      10. unpow2 46.7%

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

      11. associate-*r/ 46.7%

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

      12. *-commutative 46.7%

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

      13. associate-/l* 45.3%

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

   7. Simplified 45.3%

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

   if -5.00000000000023e-311 < l

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   6. Applied egg-rr 78.6%

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

      1. unpow1 78.6%

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

      2. associate-*r* 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*r/ 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*r/ 77.8%

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

      7. associate-*r* 77.8%

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

      8. associate-*r* 77.8%

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

      9. associate-/r* 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-27}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M\_m \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))))
   (if (<= M_m 1.2e-66)
     (* t_0 t_1)
     (* t_0 (* t_1 (* -0.125 (* (/ h l) (pow (* D (/ M_m d)) 2.0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double tmp;
	if (M_m <= 1.2e-66) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * (t_1 * (-0.125 * ((h / l) * pow((D * (M_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    if (m_m <= 1.2d-66) then
        tmp = t_0 * t_1
    else
        tmp = t_0 * (t_1 * ((-0.125d0) * ((h / l) * ((d_1 * (m_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (M_m <= 1.2e-66) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * (t_1 * (-0.125 * ((h / l) * Math.pow((D * (M_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if M_m <= 1.2e-66:
		tmp = t_0 * t_1
	else:
		tmp = t_0 * (t_1 * (-0.125 * ((h / l) * math.pow((D * (M_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (M_m <= 1.2e-66)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(-0.125 * Float64(Float64(h / l) * (Float64(D * Float64(M_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (M_m <= 1.2e-66)
		tmp = t_0 * t_1;
	else
		tmp = t_0 * (t_1 * (-0.125 * ((h / l) * ((D * (M_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 1.2e-66], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.20000000000000013e-66

   1. Initial program 65.5%

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

   3. Simplified 65.0%

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

      1. frac-2neg 65.0%

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

      2. sqrt-div 41.2%

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

   6. Applied egg-rr 41.2%

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

   7. Taylor expanded in d around inf 44.3%

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

   if 1.20000000000000013e-66 < M

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in M around inf 31.2%

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

      1. associate-*r* 31.3%

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

      2. times-frac 31.5%

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

      3. *-commutative 31.5%

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

      4. associate-/l* 31.6%

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

      5. unpow2 31.6%

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

      6. unpow2 31.6%

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

      7. unpow2 31.6%

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

      8. times-frac 38.9%

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

      9. swap-sqr 43.2%

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

      10. unpow2 43.2%

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

      11. associate-*r/ 43.2%

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

      12. *-commutative 43.2%

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

      13. associate-/l* 41.8%

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

   7. Simplified 41.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{-113}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-219}:\\ \;\;\;\;d \cdot \log \left(e^{t\_0}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= d -1.25e-113)
     (* d (- t_0))
     (if (<= d -1.9e-219)
       (* d (log (exp t_0)))
       (if (<= d -4e-310)
         (* d (sqrt (log1p (expm1 (* l h)))))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (d <= -1.25e-113) {
		tmp = d * -t_0;
	} else if (d <= -1.9e-219) {
		tmp = d * log(exp(t_0));
	} else if (d <= -4e-310) {
		tmp = d * sqrt(log1p(expm1((l * h))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (d <= -1.25e-113) {
		tmp = d * -t_0;
	} else if (d <= -1.9e-219) {
		tmp = d * Math.log(Math.exp(t_0));
	} else if (d <= -4e-310) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1((l * h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if d <= -1.25e-113:
		tmp = d * -t_0
	elif d <= -1.9e-219:
		tmp = d * math.log(math.exp(t_0))
	elif d <= -4e-310:
		tmp = d * math.sqrt(math.log1p(math.expm1((l * h))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= -1.25e-113)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1.9e-219)
		tmp = Float64(d * log(exp(t_0)));
	elseif (d <= -4e-310)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(l * h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -1.25e-113], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1.9e-219], N[(d * N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(l * h), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2499999999999999e-113

   1. Initial program 73.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in d around inf 4.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 57.5%

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

      12. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

   if -1.2499999999999999e-113 < d < -1.90000000000000012e-219

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in d around inf 11.5%

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

      1. add-log-exp 36.0%

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

      2. inv-pow 36.0%

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

      3. sqrt-pow1 36.0%

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

      4. metadata-eval 36.0%

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

   7. Applied egg-rr 36.0%

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

   if -1.90000000000000012e-219 < d < -3.999999999999988e-310

   1. Initial program 26.7%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in d around inf 7.3%

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

      1. add-exp-log 7.3%

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

      2. log-rec 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 7.6%

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

      3. sqr-neg 7.6%

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

      4. sqrt-unprod 6.8%

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

      5. add-sqr-sqrt 7.7%

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

      6. log1p-expm1-u 43.8%

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

      7. add-exp-log 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{h \cdot \ell}\right)\right)} \]

      8. *-commutative 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell \cdot h}\right)\right)} \]

   9. Applied egg-rr 43.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. associate-/r* 40.5%

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

      2. sqrt-div 48.3%

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

   7. Applied egg-rr 48.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-113}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-219}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-111}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-221}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -2.7e-111)
   (* d (- (pow (* l h) -0.5)))
   (if (<= d -3.6e-221)
     (* d (sqrt (log1p (expm1 (/ 1.0 (* l h))))))
     (if (<= d -4e-310)
       (* d (sqrt (log1p (expm1 (* l h)))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -2.7e-111) {
		tmp = d * -pow((l * h), -0.5);
	} else if (d <= -3.6e-221) {
		tmp = d * sqrt(log1p(expm1((1.0 / (l * h)))));
	} else if (d <= -4e-310) {
		tmp = d * sqrt(log1p(expm1((l * h))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -2.7e-111) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (d <= -3.6e-221) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1((1.0 / (l * h)))));
	} else if (d <= -4e-310) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1((l * h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -2.7e-111:
		tmp = d * -math.pow((l * h), -0.5)
	elif d <= -3.6e-221:
		tmp = d * math.sqrt(math.log1p(math.expm1((1.0 / (l * h)))))
	elif d <= -4e-310:
		tmp = d * math.sqrt(math.log1p(math.expm1((l * h))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -2.7e-111)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (d <= -3.6e-221)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(1.0 / Float64(l * h))))));
	elseif (d <= -4e-310)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(l * h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -2.7e-111], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -3.6e-221], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(l * h), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.69999999999999989e-111

   1. Initial program 73.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in d around inf 4.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 57.5%

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

      12. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

   if -2.69999999999999989e-111 < d < -3.60000000000000011e-221

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in d around inf 11.5%

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

      1. log1p-expm1-u 36.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

   7. Applied egg-rr 36.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

   if -3.60000000000000011e-221 < d < -3.999999999999988e-310

   1. Initial program 26.7%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in d around inf 7.3%

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

      1. add-exp-log 7.3%

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

      2. log-rec 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 7.6%

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

      3. sqr-neg 7.6%

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

      4. sqrt-unprod 6.8%

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

      5. add-sqr-sqrt 7.7%

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

      6. log1p-expm1-u 43.8%

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

      7. add-exp-log 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{h \cdot \ell}\right)\right)} \]

      8. *-commutative 43.8%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell \cdot h}\right)\right)} \]

   9. Applied egg-rr 43.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. associate-/r* 40.5%

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

      2. sqrt-div 48.3%

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

   7. Applied egg-rr 48.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-111}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-221}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{-56}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -3e-56)
   (* d (- (pow (* l h) -0.5)))
   (if (<= d -4e-310)
     (* d (sqrt (log1p (expm1 (* l h)))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -3e-56) {
		tmp = d * -pow((l * h), -0.5);
	} else if (d <= -4e-310) {
		tmp = d * sqrt(log1p(expm1((l * h))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -3e-56) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (d <= -4e-310) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1((l * h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -3e-56:
		tmp = d * -math.pow((l * h), -0.5)
	elif d <= -4e-310:
		tmp = d * math.sqrt(math.log1p(math.expm1((l * h))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -3e-56)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (d <= -4e-310)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(l * h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -3e-56], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(l * h), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.99999999999999989e-56

   1. Initial program 73.0%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in d around inf 4.7%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 60.7%

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

      12. neg-mul-1 60.7%

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

   8. Simplified 60.7%

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

   if -2.99999999999999989e-56 < d < -3.999999999999988e-310

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in d around inf 8.4%

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

      1. add-exp-log 8.4%

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

      2. log-rec 8.4%

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

   7. Applied egg-rr 8.4%

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

      1. add-sqr-sqrt 4.8%

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

      2. sqrt-unprod 10.1%

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

      3. sqr-neg 10.1%

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

      4. sqrt-unprod 5.3%

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

      5. add-sqr-sqrt 6.2%

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

      6. log1p-expm1-u 30.0%

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

      7. add-exp-log 30.0%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{h \cdot \ell}\right)\right)} \]

      8. *-commutative 30.0%

         \[\leadsto d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell \cdot h}\right)\right)} \]

   9. Applied egg-rr 30.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. associate-/r* 40.5%

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

      2. sqrt-div 48.3%

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

   7. Applied egg-rr 48.3%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{-56}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-174}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -7.6e-174)
   (* d (- (pow (* l h) -0.5)))
   (if (<= d -4e-310)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.6e-174) {
		tmp = d * -pow((l * h), -0.5);
	} else if (d <= -4e-310) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-7.6d-174)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else if (d <= (-4d-310)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -7.6e-174) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (d <= -4e-310) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -7.6e-174:
		tmp = d * -math.pow((l * h), -0.5)
	elif d <= -4e-310:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -7.6e-174)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (d <= -4e-310)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -7.6e-174)
		tmp = d * -((l * h) ^ -0.5);
	elseif (d <= -4e-310)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -7.6e-174], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.60000000000000042e-174

   1. Initial program 69.8%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in d around inf 4.5%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 53.9%

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

      12. neg-mul-1 53.9%

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

   8. Simplified 53.9%

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

   if -7.60000000000000042e-174 < d < -3.999999999999988e-310

   1. Initial program 43.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in d around inf 11.4%

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

      1. add-exp-log 11.4%

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

      2. log-rec 11.4%

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

   7. Applied egg-rr 11.4%

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

      1. exp-neg 11.4%

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

      2. add-exp-log 11.4%

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

      3. inv-pow 11.4%

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

      4. metadata-eval 11.4%

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

      5. pow-prod-up 11.4%

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

      6. sqrt-unprod 11.4%

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

      7. add-sqr-sqrt 11.4%

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

      8. sqr-pow 11.4%

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

      9. pow-prod-down 29.3%

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

      10. pow2 29.3%

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

      11. *-commutative 29.3%

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

      12. metadata-eval 29.3%

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

   9. Applied egg-rr 29.3%

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

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. associate-/r* 40.5%

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

      2. sqrt-div 48.3%

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

   7. Applied egg-rr 48.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-174}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.5% accurate, 1.6× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 1e-176)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1e-176) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 1d-176) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1e-176) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 1e-176:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 1e-176)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 1e-176)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 1e-176], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1e-176

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 38.5%

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

      6. neg-mul-1 38.5%

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

   7. Simplified 38.5%

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

   if 1e-176 < d

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in d around inf 49.6%

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

      1. *-un-lft-identity 49.6%

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

      2. inv-pow 49.6%

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

      3. sqrt-pow1 49.7%

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

      4. metadata-eval 49.7%

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

   7. Applied egg-rr 49.7%

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

      1. *-lft-identity 49.7%

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

   9. Simplified 49.7%

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

       1. *-commutative 49.7%

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

       2. unpow-prod-down 59.7%

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

   11. Applied egg-rr 59.7%

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

4. Final simplification 46.1%

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

Alternative 21: 45.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8.5 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 8.5e-176)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 8.5e-176) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 8.5d-176) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 8.5e-176) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 8.5e-176:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 8.5e-176)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 8.5e-176)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 8.5e-176], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.5e-176

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 38.5%

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

      6. neg-mul-1 38.5%

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

   7. Simplified 38.5%

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

   if 8.5e-176 < d

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in d around inf 49.6%

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

      1. associate-/r* 49.5%

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

      2. sqrt-div 59.8%

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

   7. Applied egg-rr 59.8%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.5 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 9e-177) (* (- d) (sqrt (/ (/ 1.0 h) l))) (* d (pow (* l h) -0.5))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 9e-177) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 9d-177) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 9e-177) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 9e-177:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 9e-177)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 9e-177)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 9e-177], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.0000000000000007e-177

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 38.5%

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

      6. neg-mul-1 38.5%

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

   7. Simplified 38.5%

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

   if 9.0000000000000007e-177 < d

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in d around inf 49.6%

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

      1. *-un-lft-identity 49.6%

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

      2. inv-pow 49.6%

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

      3. sqrt-pow1 49.7%

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

      4. metadata-eval 49.7%

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

   7. Applied egg-rr 49.7%

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

      1. *-lft-identity 49.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 49.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l 1.4e-224) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= 1.4e-224) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= 1.4d-224) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= 1.4e-224) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= 1.4e-224:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= 1.4e-224)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= 1.4e-224)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, 1.4e-224], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3999999999999999e-224

   1. Initial program 64.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 7.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 40.8%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 40.8%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 40.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.3999999999999999e-224 < l

   1. Initial program 65.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 44.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 44.2%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 44.2%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 44.2%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 44.2%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 44.2%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 44.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 44.2%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (pow (* l h) -0.5)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.pow((l * h), -0.5)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 64.5%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 22.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 22.4%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. inv-pow 22.4%

      \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

   3. sqrt-pow1 22.4%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   4. metadata-eval 22.4%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 22.4%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 22.4%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 22.4%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 22.4%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 25: 4.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\ell \cdot h} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 64.5%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 22.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 22.4%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. inv-pow 22.4%

      \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

   3. sqrt-pow1 22.4%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   4. metadata-eval 22.4%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 22.4%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 22.4%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 22.4%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 22.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    2. sqrt-unprod 22.4%

       \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    3. pow-prod-up 22.4%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

    4. metadata-eval 22.4%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

    5. inv-pow 22.4%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

    6. add-exp-log 21.6%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

    7. exp-neg 21.6%

       \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

    8. add-sqr-sqrt 12.2%

       \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{-\log \left(h \cdot \ell\right)} \cdot \sqrt{-\log \left(h \cdot \ell\right)}}}} \]

    9. sqrt-unprod 15.3%

       \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{\left(-\log \left(h \cdot \ell\right)\right) \cdot \left(-\log \left(h \cdot \ell\right)\right)}}}} \]

    10. sqr-neg 15.3%

        \[\leadsto d \cdot \sqrt{e^{\sqrt{\color{blue}{\log \left(h \cdot \ell\right) \cdot \log \left(h \cdot \ell\right)}}}} \]

    11. sqrt-unprod 2.9%

        \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{\log \left(h \cdot \ell\right)} \cdot \sqrt{\log \left(h \cdot \ell\right)}}}} \]

    12. add-sqr-sqrt 4.1%

        \[\leadsto d \cdot \sqrt{e^{\color{blue}{\log \left(h \cdot \ell\right)}}} \]

    13. add-exp-log 4.1%

        \[\leadsto d \cdot \sqrt{\color{blue}{h \cdot \ell}} \]

    14. *-un-lft-identity 4.1%

        \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \left(h \cdot \ell\right)}} \]

    15. sqrt-prod 4.1%

        \[\leadsto d \cdot \color{blue}{\left(\sqrt{1} \cdot \sqrt{h \cdot \ell}\right)} \]

    16. metadata-eval 4.1%

        \[\leadsto d \cdot \left(\color{blue}{1} \cdot \sqrt{h \cdot \ell}\right) \]

    17. *-commutative 4.1%

        \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{\ell \cdot h}}\right) \]

11. Applied egg-rr 4.1%

    \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\ell \cdot h}\right)} \]
12. Step-by-step derivation

    1. *-lft-identity 4.1%

       \[\leadsto d \cdot \color{blue}{\sqrt{\ell \cdot h}} \]

13. Simplified 4.1%

    \[\leadsto d \cdot \color{blue}{\sqrt{\ell \cdot h}} \]

14. Final simplification 4.1%

    \[\leadsto d \cdot \sqrt{\ell \cdot h} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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)))))