Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 77.4%

Time: 26.6s

Alternatives: 19

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: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{-h}\\ t_4 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_2}{t_3} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-70}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{t_3}{t_4}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{t_4} \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-159}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d}{{\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}}}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+112}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + -0.125 \cdot {\left(\frac{\sqrt{h} \cdot \left(M \cdot D\right)}{d \cdot \sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (- d)))
        (t_3 (sqrt (- h)))
        (t_4 (sqrt (- l))))
   (if (<= d -3.8e+46)
     (*
      (/ t_2 t_3)
      (* t_0 (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
     (if (<= d -9.6e-70)
       (*
        t_1
        (*
         t_0
         (- 1.0 (* 0.5 (pow (* (* 0.5 M) (* (/ D d) (/ t_3 t_4))) 2.0)))))
       (if (<= d -4e-279)
         (*
          t_1
          (*
           (/ t_2 t_4)
           (- 1.0 (* 0.5 (pow (* (/ (* D (* 0.5 M)) d) (sqrt (/ h l))) 2.0)))))
         (if (<= d 2.15e-159)
           (*
            -0.125
            (/ (pow (* M D) 2.0) (/ d (pow (/ (cbrt (sqrt h)) (sqrt l)) 3.0))))
           (if (<= d 1.35e+112)
             (*
              (* t_0 t_1)
              (+
               1.0
               (* -0.125 (pow (/ (* (sqrt h) (* M D)) (* d (sqrt l))) 2.0))))
             (*
              (/ d (* (sqrt h) (sqrt l)))
              (-
               1.0
               (* (/ h l) (* 0.5 (pow (* (/ D d) (* 0.5 M)) 2.0))))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double t_3 = sqrt(-h);
	double t_4 = sqrt(-l);
	double tmp;
	if (d <= -3.8e+46) {
		tmp = (t_2 / t_3) * (t_0 * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (d <= -9.6e-70) {
		tmp = t_1 * (t_0 * (1.0 - (0.5 * pow(((0.5 * M) * ((D / d) * (t_3 / t_4))), 2.0))));
	} else if (d <= -4e-279) {
		tmp = t_1 * ((t_2 / t_4) * (1.0 - (0.5 * pow((((D * (0.5 * M)) / d) * sqrt((h / l))), 2.0))));
	} else if (d <= 2.15e-159) {
		tmp = -0.125 * (pow((M * D), 2.0) / (d / pow((cbrt(sqrt(h)) / sqrt(l)), 3.0)));
	} else if (d <= 1.35e+112) {
		tmp = (t_0 * t_1) * (1.0 + (-0.125 * pow(((sqrt(h) * (M * D)) / (d * sqrt(l))), 2.0)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double t_3 = Math.sqrt(-h);
	double t_4 = Math.sqrt(-l);
	double tmp;
	if (d <= -3.8e+46) {
		tmp = (t_2 / t_3) * (t_0 * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (d <= -9.6e-70) {
		tmp = t_1 * (t_0 * (1.0 - (0.5 * Math.pow(((0.5 * M) * ((D / d) * (t_3 / t_4))), 2.0))));
	} else if (d <= -4e-279) {
		tmp = t_1 * ((t_2 / t_4) * (1.0 - (0.5 * Math.pow((((D * (0.5 * M)) / d) * Math.sqrt((h / l))), 2.0))));
	} else if (d <= 2.15e-159) {
		tmp = -0.125 * (Math.pow((M * D), 2.0) / (d / Math.pow((Math.cbrt(Math.sqrt(h)) / Math.sqrt(l)), 3.0)));
	} else if (d <= 1.35e+112) {
		tmp = (t_0 * t_1) * (1.0 + (-0.125 * Math.pow(((Math.sqrt(h) * (M * D)) / (d * Math.sqrt(l))), 2.0)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * Math.pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	t_3 = sqrt(Float64(-h))
	t_4 = sqrt(Float64(-l))
	tmp = 0.0
	if (d <= -3.8e+46)
		tmp = Float64(Float64(t_2 / t_3) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	elseif (d <= -9.6e-70)
		tmp = Float64(t_1 * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(Float64(0.5 * M) * Float64(Float64(D / d) * Float64(t_3 / t_4))) ^ 2.0)))));
	elseif (d <= -4e-279)
		tmp = Float64(t_1 * Float64(Float64(t_2 / t_4) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D * Float64(0.5 * M)) / d) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (d <= 2.15e-159)
		tmp = Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / Float64(d / (Float64(cbrt(sqrt(h)) / sqrt(l)) ^ 3.0))));
	elseif (d <= 1.35e+112)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 + Float64(-0.125 * (Float64(Float64(sqrt(h) * Float64(M * D)) / Float64(d * sqrt(l))) ^ 2.0))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-h)], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-l)], $MachinePrecision]}, If[LessEqual[d, -3.8e+46], N[(N[(t$95$2 / t$95$3), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -9.6e-70], N[(t$95$1 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(t$95$3 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-279], N[(t$95$1 * N[(N[(t$95$2 / t$95$4), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e-159], N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(d / N[Power[N[(N[Power[N[Sqrt[h], $MachinePrecision], 1/3], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e+112], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[Power[N[(N[(N[Sqrt[h], $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -3.7999999999999999e46

   1. Initial program 78.1%

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

   3. Simplified 78.1%

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

      1. frac-2neg 78.1%

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

      2. sqrt-div 87.8%

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

   5. Applied egg-rr 87.8%

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

   if -3.7999999999999999e46 < d < -9.6000000000000005e-70

   1. Initial program 79.3%

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

   3. Simplified 79.2%

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

      1. add-sqr-sqrt 79.2%

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

      2. pow2 79.2%

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

      3. sqrt-prod 79.2%

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

      4. unpow2 79.2%

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

      5. sqrt-prod 37.5%

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

      6. add-sqr-sqrt 87.5%

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

      7. div-inv 87.5%

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

      8. metadata-eval 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-*l* 87.6%

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

   7. Simplified 87.6%

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

      1. frac-2neg 87.6%

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

      2. sqrt-div 98.8%

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

   9. Applied egg-rr 98.8%

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

   if -9.6000000000000005e-70 < d < -4.00000000000000022e-279

   1. Initial program 36.6%

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

   3. Simplified 36.4%

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

      1. add-sqr-sqrt 36.4%

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

      2. pow2 36.4%

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

      3. sqrt-prod 36.4%

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

      4. unpow2 36.4%

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

      5. sqrt-prod 24.4%

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

      6. add-sqr-sqrt 39.9%

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

      7. div-inv 39.9%

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

      8. metadata-eval 39.9%

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

   5. Applied egg-rr 39.9%

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

      1. associate-*r/ 43.4%

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

   7. Applied egg-rr 43.4%

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

      1. frac-2neg 43.4%

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

      2. sqrt-div 60.5%

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

   9. Applied egg-rr 60.5%

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

   if -4.00000000000000022e-279 < d < 2.15e-159

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

      1. fma-udef 20.1%

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

      2. *-commutative 20.1%

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

      3. *-un-lft-identity 20.1%

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

      4. times-frac 20.1%

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

      5. metadata-eval 20.1%

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

      6. *-commutative 20.1%

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

   5. Applied egg-rr 20.1%

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

   6. Taylor expanded in d around 0 29.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 28.8%

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

      2. unpow2 28.8%

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

      3. unpow2 28.8%

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

      4. swap-sqr 36.5%

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

      5. unpow2 36.5%

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

      6. associate-/l* 37.0%

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

   8. Simplified 37.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 37.1%

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

      2. pow3 37.1%

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

      3. sqrt-div 38.6%

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

      4. cbrt-div 38.7%

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

      5. unpow3 38.7%

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

      6. sqrt-prod 57.6%

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

      7. sqrt-unprod 64.8%

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

      8. add-cbrt-cube 67.5%

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

   10. Applied egg-rr 67.5%

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

   if 2.15e-159 < d < 1.3500000000000001e112

   1. Initial program 73.3%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in M around 0 62.3%

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

      1. add-sqr-sqrt 62.3%

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

      2. sqrt-div 62.3%

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

      3. sqrt-prod 62.3%

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

      4. unpow2 62.3%

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

      5. sqrt-prod 29.8%

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

      6. add-sqr-sqrt 45.4%

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

      7. sqrt-prod 45.4%

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

      8. unpow2 45.4%

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

      9. sqrt-prod 20.8%

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

      10. add-sqr-sqrt 44.3%

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

      11. sqrt-prod 44.3%

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

      12. unpow2 44.3%

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

      13. sqrt-prod 44.3%

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

      14. add-sqr-sqrt 44.3%

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

      15. sqrt-div 44.3%

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

   6. Applied egg-rr 85.8%

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

      1. unpow2 85.8%

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

      2. associate-*r* 87.0%

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

   8. Simplified 87.0%

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

   if 1.3500000000000001e112 < d

   1. Initial program 71.0%

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

      1. pow1 71.0%

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

   3. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)}^{1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{\sqrt{-h}}{\sqrt{-\ell}}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-159}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d}{{\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}}}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+112}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.125 \cdot {\left(\frac{\sqrt{h} \cdot \left(M \cdot D\right)}{d \cdot \sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d}{{\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (fabs (* d (pow (* h l) -0.5))))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -4e+15)
     (*
      t_3
      (*
       t_0
       (- 1.0 (* 0.5 (pow (* (/ (* D (* 0.5 M)) d) (sqrt (/ h l))) 2.0)))))
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+253)
         (*
          (* t_0 t_3)
          (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
         (if (<= t_1 INFINITY)
           t_2
           (*
            -0.125
            (/
             (pow (* M D) 2.0)
             (/ d (pow (/ (cbrt (sqrt h)) (sqrt l)) 3.0))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = fabs((d * pow((h * l), -0.5)));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * pow((((D * (0.5 * M)) / d) * sqrt((h / l))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.125 * (pow((M * D), 2.0) / (d / pow((cbrt(sqrt(h)) / sqrt(l)), 3.0)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.abs((d * Math.pow((h * l), -0.5)));
	double t_3 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * Math.pow((((D * (0.5 * M)) / d) * Math.sqrt((h / l))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.125 * (Math.pow((M * D), 2.0) / (d / Math.pow((Math.cbrt(Math.sqrt(h)) / Math.sqrt(l)), 3.0)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = abs(Float64(d * (Float64(h * l) ^ -0.5)))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D * Float64(0.5 * M)) / d) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = Float64(Float64(t_0 * t_3) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / Float64(d / (Float64(cbrt(sqrt(h)) / sqrt(l)) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$3 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+253], N[(N[(t$95$0 * t$95$3), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(d / N[Power[N[(N[Power[N[Sqrt[h], $MachinePrecision], 1/3], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. pow2 79.7%

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

      3. sqrt-prod 79.6%

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

      4. unpow2 79.6%

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

      5. sqrt-prod 44.1%

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

      6. add-sqr-sqrt 87.5%

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

      7. div-inv 87.5%

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

      8. metadata-eval 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-*r/ 87.4%

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

   7. Applied egg-rr 87.4%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-unprod 55.5%

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

      3. pow2 55.5%

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

      4. inv-pow 55.5%

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

      5. sqrt-pow1 55.6%

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

      6. metadata-eval 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow2 55.6%

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

      2. rem-sqrt-square 92.5%

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

   8. Simplified 92.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. times-frac 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around 0 12.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. swap-sqr 17.0%

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

      5. unpow2 17.0%

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

      6. associate-/l* 17.0%

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

   8. Simplified 17.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 17.0%

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

      2. pow3 17.0%

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

      3. sqrt-div 16.2%

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

      4. cbrt-div 16.2%

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

      5. unpow3 16.2%

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

      6. sqrt-prod 29.3%

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

      7. sqrt-unprod 35.0%

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

      8. add-cbrt-cube 37.0%

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

   10. Applied egg-rr 37.0%

       \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\color{blue}{{\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d}{{\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}}}\\ \end{array} \]

Alternative 3: 76.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 + \frac{h \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(t_3 \cdot t_0\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (fabs (* d (pow (* h l) -0.5))))
        (t_3 (sqrt (/ d l))))
   (if (<= t_1 -4e+15)
     (*
      t_3
      (* t_0 (+ 1.0 (/ (* h (* (pow (* (/ D d) (* 0.5 M)) 2.0) -0.5)) l))))
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+253)
         (*
          (* t_3 t_0)
          (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
         (if (<= t_1 INFINITY)
           t_2
           (*
            -0.125
            (pow (/ (* M D) (sqrt (/ d (/ (sqrt h) (pow l 1.5))))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = fabs((d * pow((h * l), -0.5)));
	double t_3 = sqrt((d / l));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 + ((h * (pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_3 * t_0) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.125 * pow(((M * D) / sqrt((d / (sqrt(h) / pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.abs((d * Math.pow((h * l), -0.5)));
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 + ((h * (Math.pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_3 * t_0) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.125 * Math.pow(((M * D) / Math.sqrt((d / (Math.sqrt(h) / Math.pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_2 = math.fabs((d * math.pow((h * l), -0.5)))
	t_3 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_3 * (t_0 * (1.0 + ((h * (math.pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)))
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2e+253:
		tmp = (t_3 * t_0) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = -0.125 * math.pow(((M * D) / math.sqrt((d / (math.sqrt(h) / math.pow(l, 1.5))))), 2.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = abs(Float64(d * (Float64(h * l) ^ -0.5)))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0) * -0.5)) / l))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = Float64(Float64(t_3 * t_0) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.125 * (Float64(Float64(M * D) / sqrt(Float64(d / Float64(sqrt(h) / (l ^ 1.5))))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	t_2 = abs((d * ((h * l) ^ -0.5)));
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_3 * (t_0 * (1.0 + ((h * ((((D / d) * (0.5 * M)) ^ 2.0) * -0.5)) / l)));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = (t_3 * t_0) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = -0.125 * (((M * D) / sqrt((d / (sqrt(h) / (l ^ 1.5))))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$3 * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+253], N[(N[(t$95$3 * t$95$0), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-0.125 * N[Power[N[(N[(M * D), $MachinePrecision] / N[Sqrt[N[(d / N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. associate-*l/ 79.7%

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

      2. add-sqr-sqrt 38.8%

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

      3. add-sqr-sqrt 79.7%

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

      4. div-inv 79.7%

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

      5. metadata-eval 79.7%

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

   5. Applied egg-rr 79.7%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-unprod 55.5%

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

      3. pow2 55.5%

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

      4. inv-pow 55.5%

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

      5. sqrt-pow1 55.6%

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

      6. metadata-eval 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow2 55.6%

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

      2. rem-sqrt-square 92.5%

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

   8. Simplified 92.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. times-frac 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around 0 12.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. swap-sqr 17.0%

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

      5. unpow2 17.0%

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

      6. associate-/l* 17.0%

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

   8. Simplified 17.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 16.6%

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

      2. pow2 16.6%

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

      3. sqrt-div 22.3%

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

      4. unpow2 22.3%

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

      5. sqrt-prod 6.6%

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

      6. add-sqr-sqrt 22.7%

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

      7. *-commutative 22.7%

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

      8. sqrt-div 16.5%

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

      9. sqrt-pow1 35.6%

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

      10. metadata-eval 35.6%

          \[\leadsto -0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}}}}\right)}^{2} \]

   10. Applied egg-rr 35.6%

       \[\leadsto -0.125 \cdot \color{blue}{{\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \]

Alternative 4: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot M\right) \cdot \left(\frac{D}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (fabs (* d (pow (* h l) -0.5))))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -4e+15)
     (*
      t_3
      (*
       t_0
       (- 1.0 (* 0.5 (pow (* (* 0.5 M) (* (/ D d) (sqrt (/ h l)))) 2.0)))))
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+253)
         (*
          (* t_0 t_3)
          (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
         (if (<= t_1 INFINITY)
           t_2
           (*
            -0.125
            (pow (/ (* M D) (sqrt (/ d (/ (sqrt h) (pow l 1.5))))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = fabs((d * pow((h * l), -0.5)));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * pow(((0.5 * M) * ((D / d) * sqrt((h / l)))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.125 * pow(((M * D) / sqrt((d / (sqrt(h) / pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.abs((d * Math.pow((h * l), -0.5)));
	double t_3 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * Math.pow(((0.5 * M) * ((D / d) * Math.sqrt((h / l)))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.125 * Math.pow(((M * D) / Math.sqrt((d / (Math.sqrt(h) / Math.pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_2 = math.fabs((d * math.pow((h * l), -0.5)))
	t_3 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_3 * (t_0 * (1.0 - (0.5 * math.pow(((0.5 * M) * ((D / d) * math.sqrt((h / l)))), 2.0))))
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2e+253:
		tmp = (t_0 * t_3) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = -0.125 * math.pow(((M * D) / math.sqrt((d / (math.sqrt(h) / math.pow(l, 1.5))))), 2.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = abs(Float64(d * (Float64(h * l) ^ -0.5)))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(Float64(0.5 * M) * Float64(Float64(D / d) * sqrt(Float64(h / l)))) ^ 2.0)))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = Float64(Float64(t_0 * t_3) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.125 * (Float64(Float64(M * D) / sqrt(Float64(d / Float64(sqrt(h) / (l ^ 1.5))))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	t_2 = abs((d * ((h * l) ^ -0.5)));
	t_3 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_3 * (t_0 * (1.0 - (0.5 * (((0.5 * M) * ((D / d) * sqrt((h / l)))) ^ 2.0))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = (t_0 * t_3) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = -0.125 * (((M * D) / sqrt((d / (sqrt(h) / (l ^ 1.5))))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$3 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+253], N[(N[(t$95$0 * t$95$3), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-0.125 * N[Power[N[(N[(M * D), $MachinePrecision] / N[Sqrt[N[(d / N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. pow2 79.7%

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

      3. sqrt-prod 79.6%

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

      4. unpow2 79.6%

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

      5. sqrt-prod 44.1%

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

      6. add-sqr-sqrt 87.5%

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

      7. div-inv 87.5%

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

      8. metadata-eval 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-*l* 86.4%

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

   7. Simplified 86.4%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-unprod 55.5%

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

      3. pow2 55.5%

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

      4. inv-pow 55.5%

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

      5. sqrt-pow1 55.6%

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

      6. metadata-eval 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow2 55.6%

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

      2. rem-sqrt-square 92.5%

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

   8. Simplified 92.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. times-frac 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around 0 12.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. swap-sqr 17.0%

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

      5. unpow2 17.0%

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

      6. associate-/l* 17.0%

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

   8. Simplified 17.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 16.6%

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

      2. pow2 16.6%

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

      3. sqrt-div 22.3%

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

      4. unpow2 22.3%

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

      5. sqrt-prod 6.6%

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

      6. add-sqr-sqrt 22.7%

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

      7. *-commutative 22.7%

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

      8. sqrt-div 16.5%

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

      9. sqrt-pow1 35.6%

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

      10. metadata-eval 35.6%

          \[\leadsto -0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}}}}\right)}^{2} \]

   10. Applied egg-rr 35.6%

       \[\leadsto -0.125 \cdot \color{blue}{{\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot M\right) \cdot \left(\frac{D}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \]

Alternative 5: 77.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (fabs (* d (pow (* h l) -0.5))))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -4e+15)
     (*
      t_3
      (*
       t_0
       (- 1.0 (* 0.5 (pow (* (sqrt (/ h l)) (* (/ D d) (* 0.5 M))) 2.0)))))
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+253)
         (*
          (* t_0 t_3)
          (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
         (if (<= t_1 INFINITY)
           t_2
           (*
            -0.125
            (pow (/ (* M D) (sqrt (/ d (/ (sqrt h) (pow l 1.5))))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = fabs((d * pow((h * l), -0.5)));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * pow((sqrt((h / l)) * ((D / d) * (0.5 * M))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.125 * pow(((M * D) / sqrt((d / (sqrt(h) / pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.abs((d * Math.pow((h * l), -0.5)));
	double t_3 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * Math.pow((Math.sqrt((h / l)) * ((D / d) * (0.5 * M))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.125 * Math.pow(((M * D) / Math.sqrt((d / (Math.sqrt(h) / Math.pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_2 = math.fabs((d * math.pow((h * l), -0.5)))
	t_3 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_3 * (t_0 * (1.0 - (0.5 * math.pow((math.sqrt((h / l)) * ((D / d) * (0.5 * M))), 2.0))))
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2e+253:
		tmp = (t_0 * t_3) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = -0.125 * math.pow(((M * D) / math.sqrt((d / (math.sqrt(h) / math.pow(l, 1.5))))), 2.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = abs(Float64(d * (Float64(h * l) ^ -0.5)))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(sqrt(Float64(h / l)) * Float64(Float64(D / d) * Float64(0.5 * M))) ^ 2.0)))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = Float64(Float64(t_0 * t_3) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.125 * (Float64(Float64(M * D) / sqrt(Float64(d / Float64(sqrt(h) / (l ^ 1.5))))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	t_2 = abs((d * ((h * l) ^ -0.5)));
	t_3 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_3 * (t_0 * (1.0 - (0.5 * ((sqrt((h / l)) * ((D / d) * (0.5 * M))) ^ 2.0))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = (t_0 * t_3) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = -0.125 * (((M * D) / sqrt((d / (sqrt(h) / (l ^ 1.5))))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$3 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+253], N[(N[(t$95$0 * t$95$3), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-0.125 * N[Power[N[(N[(M * D), $MachinePrecision] / N[Sqrt[N[(d / N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. pow2 79.7%

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

      3. sqrt-prod 79.6%

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

      4. unpow2 79.6%

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

      5. sqrt-prod 44.1%

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

      6. add-sqr-sqrt 87.5%

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

      7. div-inv 87.5%

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

      8. metadata-eval 87.5%

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

   5. Applied egg-rr 87.5%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-unprod 55.5%

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

      3. pow2 55.5%

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

      4. inv-pow 55.5%

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

      5. sqrt-pow1 55.6%

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

      6. metadata-eval 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow2 55.6%

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

      2. rem-sqrt-square 92.5%

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

   8. Simplified 92.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. times-frac 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around 0 12.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. swap-sqr 17.0%

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

      5. unpow2 17.0%

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

      6. associate-/l* 17.0%

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

   8. Simplified 17.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 16.6%

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

      2. pow2 16.6%

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

      3. sqrt-div 22.3%

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

      4. unpow2 22.3%

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

      5. sqrt-prod 6.6%

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

      6. add-sqr-sqrt 22.7%

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

      7. *-commutative 22.7%

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

      8. sqrt-div 16.5%

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

      9. sqrt-pow1 35.6%

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

      10. metadata-eval 35.6%

          \[\leadsto -0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}}}}\right)}^{2} \]

   10. Applied egg-rr 35.6%

       \[\leadsto -0.125 \cdot \color{blue}{{\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \]

Alternative 6: 77.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (fabs (* d (pow (* h l) -0.5))))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -4e+15)
     (*
      t_3
      (*
       t_0
       (- 1.0 (* 0.5 (pow (* (/ (* D (* 0.5 M)) d) (sqrt (/ h l))) 2.0)))))
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 2e+253)
         (*
          (* t_0 t_3)
          (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
         (if (<= t_1 INFINITY)
           t_2
           (*
            -0.125
            (pow (/ (* M D) (sqrt (/ d (/ (sqrt h) (pow l 1.5))))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = fabs((d * pow((h * l), -0.5)));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * pow((((D * (0.5 * M)) / d) * sqrt((h / l))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.125 * pow(((M * D) / sqrt((d / (sqrt(h) / pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.abs((d * Math.pow((h * l), -0.5)));
	double t_3 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_3 * (t_0 * (1.0 - (0.5 * Math.pow((((D * (0.5 * M)) / d) * Math.sqrt((h / l))), 2.0))));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+253) {
		tmp = (t_0 * t_3) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.125 * Math.pow(((M * D) / Math.sqrt((d / (Math.sqrt(h) / Math.pow(l, 1.5))))), 2.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_2 = math.fabs((d * math.pow((h * l), -0.5)))
	t_3 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_3 * (t_0 * (1.0 - (0.5 * math.pow((((D * (0.5 * M)) / d) * math.sqrt((h / l))), 2.0))))
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2e+253:
		tmp = (t_0 * t_3) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = -0.125 * math.pow(((M * D) / math.sqrt((d / (math.sqrt(h) / math.pow(l, 1.5))))), 2.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = abs(Float64(d * (Float64(h * l) ^ -0.5)))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D * Float64(0.5 * M)) / d) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = Float64(Float64(t_0 * t_3) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.125 * (Float64(Float64(M * D) / sqrt(Float64(d / Float64(sqrt(h) / (l ^ 1.5))))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	t_2 = abs((d * ((h * l) ^ -0.5)));
	t_3 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_3 * (t_0 * (1.0 - (0.5 * ((((D * (0.5 * M)) / d) * sqrt((h / l))) ^ 2.0))));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+253)
		tmp = (t_0 * t_3) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = -0.125 * (((M * D) / sqrt((d / (sqrt(h) / (l ^ 1.5))))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$3 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+253], N[(N[(t$95$0 * t$95$3), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-0.125 * N[Power[N[(N[(M * D), $MachinePrecision] / N[Sqrt[N[(d / N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. pow2 79.7%

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

      3. sqrt-prod 79.6%

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

      4. unpow2 79.6%

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

      5. sqrt-prod 44.1%

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

      6. add-sqr-sqrt 87.5%

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

      7. div-inv 87.5%

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

      8. metadata-eval 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-*r/ 87.4%

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

   7. Applied egg-rr 87.4%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-unprod 55.5%

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

      3. pow2 55.5%

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

      4. inv-pow 55.5%

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

      5. sqrt-pow1 55.6%

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

      6. metadata-eval 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow2 55.6%

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

      2. rem-sqrt-square 92.5%

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

   8. Simplified 92.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. times-frac 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around 0 12.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. swap-sqr 17.0%

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

      5. unpow2 17.0%

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

      6. associate-/l* 17.0%

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

   8. Simplified 17.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 16.6%

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

      2. pow2 16.6%

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

      3. sqrt-div 22.3%

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

      4. unpow2 22.3%

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

      5. sqrt-prod 6.6%

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

      6. add-sqr-sqrt 22.7%

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

      7. *-commutative 22.7%

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

      8. sqrt-div 16.5%

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

      9. sqrt-pow1 35.6%

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

      10. metadata-eval 35.6%

          \[\leadsto -0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}}}}\right)}^{2} \]

   10. Applied egg-rr 35.6%

       \[\leadsto -0.125 \cdot \color{blue}{{\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot {\left(\frac{M \cdot D}{\sqrt{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}}\right)}^{2}\\ \end{array} \]

Alternative 7: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(1 + \frac{h \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2 \cdot 10^{+253}\right):\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_2 (sqrt (/ d l))))
   (if (<= t_1 -4e+15)
     (*
      t_2
      (* t_0 (+ 1.0 (/ (* h (* (pow (* (/ D d) (* 0.5 M)) 2.0) -0.5)) l))))
     (if (or (<= t_1 0.0) (not (<= t_1 2e+253)))
       (fabs (* d (pow (* h l) -0.5)))
       (*
        (* t_2 t_0)
        (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_2 * (t_0 * (1.0 + ((h * (pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)));
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+253)) {
		tmp = fabs((d * pow((h * l), -0.5)));
	} else {
		tmp = (t_2 * t_0) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    t_2 = sqrt((d / l))
    if (t_1 <= (-4d+15)) then
        tmp = t_2 * (t_0 * (1.0d0 + ((h * ((((d_1 / d) * (0.5d0 * m)) ** 2.0d0) * (-0.5d0))) / l)))
    else if ((t_1 <= 0.0d0) .or. (.not. (t_1 <= 2d+253))) then
        tmp = abs((d * ((h * l) ** (-0.5d0))))
    else
        tmp = (t_2 * t_0) * (1.0d0 + (((d_1 * ((-0.5d0) * (m / d))) ** 2.0d0) * ((h / l) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_2 * (t_0 * (1.0 + ((h * (Math.pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)));
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+253)) {
		tmp = Math.abs((d * Math.pow((h * l), -0.5)));
	} else {
		tmp = (t_2 * t_0) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_2 * (t_0 * (1.0 + ((h * (math.pow(((D / d) * (0.5 * M)), 2.0) * -0.5)) / l)))
	elif (t_1 <= 0.0) or not (t_1 <= 2e+253):
		tmp = math.fabs((d * math.pow((h * l), -0.5)))
	else:
		tmp = (t_2 * t_0) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = Float64(t_2 * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0) * -0.5)) / l))));
	elseif ((t_1 <= 0.0) || !(t_1 <= 2e+253))
		tmp = abs(Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(t_2 * t_0) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_2 * (t_0 * (1.0 + ((h * ((((D / d) * (0.5 * M)) ^ 2.0) * -0.5)) / l)));
	elseif ((t_1 <= 0.0) || ~((t_1 <= 2e+253)))
		tmp = abs((d * ((h * l) ^ -0.5)));
	else
		tmp = (t_2 * t_0) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], N[(t$95$2 * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 2e+253]], $MachinePrecision]], N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4e15

   1. Initial program 78.5%

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

   3. Simplified 79.7%

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

      1. associate-*l/ 79.7%

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

      2. add-sqr-sqrt 38.8%

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

      3. add-sqr-sqrt 79.7%

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

      4. div-inv 79.7%

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

      5. metadata-eval 79.7%

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

   5. Applied egg-rr 79.7%

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

   if -4e15 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 24.4%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in d around inf 35.3%

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

      1. add-sqr-sqrt 34.5%

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

      2. sqrt-unprod 35.9%

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

      3. pow2 35.9%

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

      4. inv-pow 35.9%

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

      5. sqrt-pow1 35.9%

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

      6. metadata-eval 35.9%

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

   6. Applied egg-rr 35.9%

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

      1. unpow2 35.9%

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

      2. rem-sqrt-square 59.9%

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

   8. Simplified 59.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.2%

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

   3. Simplified 96.8%

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

      1. fma-udef 96.8%

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

      2. *-commutative 96.8%

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

      3. *-un-lft-identity 96.8%

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

      4. times-frac 96.8%

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

      5. metadata-eval 96.8%

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

      6. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}\right):\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 8: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (fabs (* d (pow (* h l) -0.5)))
   (if (<= d 8e-99)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = fabs((d * pow((h * l), -0.5)));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-5d-310)) then
        tmp = abs((d * ((h * l) ** (-0.5d0))))
    else if (d <= 8d-99) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (((m * d_1) ** 2.0d0) / d))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = Math.abs((d * Math.pow((h * l), -0.5)));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (Math.pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = math.fabs((d * math.pow((h * l), -0.5)))
	elif d <= 8e-99:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (math.pow((M * D), 2.0) / d))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = abs(Float64(d * (Float64(h * l) ^ -0.5)));
	elseif (d <= 8e-99)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = abs((d * ((h * l) ^ -0.5)));
	elseif (d <= 8e-99)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (((M * D) ^ 2.0) / d));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, 8e-99], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   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 64.2%

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

   4. Taylor expanded in d around inf 8.7%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

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

      4. inv-pow 31.7%

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

      5. sqrt-pow1 31.7%

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

      6. metadata-eval 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. unpow2 31.7%

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

      2. rem-sqrt-square 45.4%

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

   8. Simplified 45.4%

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

   if -4.999999999999985e-310 < d < 8.0000000000000002e-99

   1. Initial program 35.3%

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

   3. Simplified 35.3%

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

      1. fma-udef 35.3%

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

      2. *-commutative 35.3%

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

      3. *-un-lft-identity 35.3%

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

      4. times-frac 35.3%

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

      5. metadata-eval 35.3%

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

      6. *-commutative 35.3%

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

   5. Applied egg-rr 35.3%

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

   6. Taylor expanded in d around 0 36.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. swap-sqr 42.5%

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

      5. unpow2 42.5%

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

      6. associate-/l* 42.9%

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

   8. Simplified 42.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 42.7%

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

      2. *-commutative 42.7%

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

      3. sqrt-div 46.2%

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

      4. sqrt-pow1 68.4%

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

      5. metadata-eval 68.4%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 68.4%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 8.0000000000000002e-99 < d

   1. Initial program 72.4%

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

   3. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 85.9%

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

      3. associate-/r* 85.0%

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

      4. associate-*r* 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \left(D \cdot -0.5\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4e-279)
   (*
    (fma (pow (* (/ M d) (* D -0.5)) 2.0) (* (/ h l) -0.5) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (if (<= d 2.15e-97)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = fma(pow(((M / d) * (D * -0.5)), 2.0), ((h / l) * -0.5), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= 2.15e-97) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4e-279)
		tmp = Float64(fma((Float64(Float64(M / d) * Float64(D * -0.5)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= 2.15e-97)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4e-279], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e-97], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000022e-279

   1. Initial program 67.4%

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

   3. Simplified 66.5%

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

      1. fma-udef 66.5%

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

      2. *-commutative 66.5%

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

      3. *-un-lft-identity 66.5%

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

      4. times-frac 66.5%

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

      5. metadata-eval 66.5%

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

      6. *-commutative 66.5%

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

   5. Applied egg-rr 66.5%

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

      1. distribute-lft-in 50.5%

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

      2. sqrt-unprod 45.1%

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

      3. associate-*r* 45.1%

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

      4. sqrt-unprod 43.3%

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

   7. Applied egg-rr 43.3%

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

      1. distribute-lft-out 60.5%

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

      2. fma-udef 60.5%

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

      3. *-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -4.00000000000000022e-279 < d < 2.15e-97

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

      1. fma-udef 32.6%

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

      2. *-commutative 32.6%

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

      3. *-un-lft-identity 32.6%

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

      4. times-frac 32.6%

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

      5. metadata-eval 32.6%

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

      6. *-commutative 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in d around 0 33.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. swap-sqr 39.4%

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

      5. unpow2 39.4%

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

      6. associate-/l* 39.8%

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. sqrt-div 42.8%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 63.2%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 2.15e-97 < d

   1. Initial program 72.4%

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

   3. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 85.9%

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

      3. associate-/r* 85.0%

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

      4. associate-*r* 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \left(D \cdot -0.5\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 10: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-98}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4e-279)
   (*
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
    (sqrt (/ d h)))
   (if (<= d 4.3e-98)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * sqrt((d / h));
	} else if (d <= 4.3e-98) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-4d-279)) then
        tmp = (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))))) * sqrt((d / h))
    else if (d <= 4.3d-98) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (((m * d_1) ** 2.0d0) / d))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = (Math.sqrt((d / l)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * Math.sqrt((d / h));
	} else if (d <= 4.3e-98) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (Math.pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -4e-279:
		tmp = (math.sqrt((d / l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * math.sqrt((d / h))
	elif d <= 4.3e-98:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (math.pow((M * D), 2.0) / d))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4e-279)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))) * sqrt(Float64(d / h)));
	elseif (d <= 4.3e-98)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -4e-279)
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))))) * sqrt((d / h));
	elseif (d <= 4.3e-98)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (((M * D) ^ 2.0) / d));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4e-279], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-98], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000022e-279

   1. Initial program 67.4%

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

   3. Simplified 67.3%

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

   if -4.00000000000000022e-279 < d < 4.29999999999999988e-98

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

      1. fma-udef 32.6%

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

      2. *-commutative 32.6%

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

      3. *-un-lft-identity 32.6%

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

      4. times-frac 32.6%

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

      5. metadata-eval 32.6%

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

      6. *-commutative 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in d around 0 33.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. swap-sqr 39.4%

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

      5. unpow2 39.4%

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

      6. associate-/l* 39.8%

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. sqrt-div 42.8%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 63.2%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 4.29999999999999988e-98 < d

   1. Initial program 72.4%

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

   3. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 85.9%

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

      3. associate-/r* 85.0%

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

      4. associate-*r* 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-98}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4e-279)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5)))))
   (if (<= d 8e-99)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5))));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-4d-279)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))))
    else if (d <= 8d-99) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (((m * d_1) ** 2.0d0) / d))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (Math.pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -4e-279:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))))
	elif d <= 8e-99:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (math.pow((M * D), 2.0) / d))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4e-279)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))));
	elseif (d <= 8e-99)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -4e-279)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5))));
	elseif (d <= 8e-99)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (((M * D) ^ 2.0) / d));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4e-279], 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[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-99], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000022e-279

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   if -4.00000000000000022e-279 < d < 8.0000000000000002e-99

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

      1. fma-udef 32.6%

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

      2. *-commutative 32.6%

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

      3. *-un-lft-identity 32.6%

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

      4. times-frac 32.6%

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

      5. metadata-eval 32.6%

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

      6. *-commutative 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in d around 0 33.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. swap-sqr 39.4%

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

      5. unpow2 39.4%

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

      6. associate-/l* 39.8%

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. sqrt-div 42.8%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 63.2%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 8.0000000000000002e-99 < d

   1. Initial program 72.4%

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

   3. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 85.9%

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

      3. associate-/r* 85.0%

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

      4. associate-*r* 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 12: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4e-279)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ M (* 2.0 (/ d D))) 2.0))))))
   (if (<= d 8e-99)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((M / (2.0 * (d / D))), 2.0)))));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-4d-279)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((m / (2.0d0 * (d / d_1))) ** 2.0d0)))))
    else if (d <= 8d-99) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (((m * d_1) ** 2.0d0) / d))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-279) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((M / (2.0 * (d / D))), 2.0)))));
	} else if (d <= 8e-99) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (Math.pow((M * D), 2.0) / d));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -4e-279:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((M / (2.0 * (d / D))), 2.0)))))
	elif d <= 8e-99:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (math.pow((M * D), 2.0) / d))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4e-279)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(M / Float64(2.0 * Float64(d / D))) ^ 2.0))))));
	elseif (d <= 8e-99)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -4e-279)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((M / (2.0 * (d / D))) ^ 2.0)))));
	elseif (d <= 8e-99)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (((M * D) ^ 2.0) / d));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4e-279], 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[(M / N[(2.0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-99], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000022e-279

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. clear-num 67.3%

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

      3. frac-times 67.4%

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

      4. *-un-lft-identity 67.4%

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

   5. Applied egg-rr 67.4%

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

   if -4.00000000000000022e-279 < d < 8.0000000000000002e-99

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

      1. fma-udef 32.6%

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

      2. *-commutative 32.6%

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

      3. *-un-lft-identity 32.6%

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

      4. times-frac 32.6%

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

      5. metadata-eval 32.6%

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

      6. *-commutative 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in d around 0 33.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. swap-sqr 39.4%

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

      5. unpow2 39.4%

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

      6. associate-/l* 39.8%

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. sqrt-div 42.8%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 63.2%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 8.0000000000000002e-99 < d

   1. Initial program 72.4%

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

   3. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 85.9%

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

      3. associate-/r* 85.0%

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

      4. associate-*r* 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-99}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 13: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := {\left(M \cdot D\right)}^{2}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-271}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{t_1}{d}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;-0.125 \cdot \left(t_1 \cdot \frac{\frac{\sqrt{h}}{d}}{{\ell}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* h l) -0.5))) (t_1 (pow (* M D) 2.0)))
   (if (<= d -5e-310)
     (fabs t_0)
     (if (<= d 3e-271)
       (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ t_1 d)))
       (if (<= d 1.4e-259)
         t_0
         (if (<= d 5.4e+55)
           (* -0.125 (* t_1 (/ (/ (sqrt h) d) (pow l 1.5))))
           (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((h * l), -0.5);
	double t_1 = pow((M * D), 2.0);
	double tmp;
	if (d <= -5e-310) {
		tmp = fabs(t_0);
	} else if (d <= 3e-271) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (t_1 / d));
	} else if (d <= 1.4e-259) {
		tmp = t_0;
	} else if (d <= 5.4e+55) {
		tmp = -0.125 * (t_1 * ((sqrt(h) / d) / pow(l, 1.5)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d * ((h * l) ** (-0.5d0))
    t_1 = (m * d_1) ** 2.0d0
    if (d <= (-5d-310)) then
        tmp = abs(t_0)
    else if (d <= 3d-271) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (t_1 / d))
    else if (d <= 1.4d-259) then
        tmp = t_0
    else if (d <= 5.4d+55) then
        tmp = (-0.125d0) * (t_1 * ((sqrt(h) / d) / (l ** 1.5d0)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((h * l), -0.5);
	double t_1 = Math.pow((M * D), 2.0);
	double tmp;
	if (d <= -5e-310) {
		tmp = Math.abs(t_0);
	} else if (d <= 3e-271) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (t_1 / d));
	} else if (d <= 1.4e-259) {
		tmp = t_0;
	} else if (d <= 5.4e+55) {
		tmp = -0.125 * (t_1 * ((Math.sqrt(h) / d) / Math.pow(l, 1.5)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.pow((h * l), -0.5)
	t_1 = math.pow((M * D), 2.0)
	tmp = 0
	if d <= -5e-310:
		tmp = math.fabs(t_0)
	elif d <= 3e-271:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (t_1 / d))
	elif d <= 1.4e-259:
		tmp = t_0
	elif d <= 5.4e+55:
		tmp = -0.125 * (t_1 * ((math.sqrt(h) / d) / math.pow(l, 1.5)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(h * l) ^ -0.5))
	t_1 = Float64(M * D) ^ 2.0
	tmp = 0.0
	if (d <= -5e-310)
		tmp = abs(t_0);
	elseif (d <= 3e-271)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(t_1 / d)));
	elseif (d <= 1.4e-259)
		tmp = t_0;
	elseif (d <= 5.4e+55)
		tmp = Float64(-0.125 * Float64(t_1 * Float64(Float64(sqrt(h) / d) / (l ^ 1.5))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * ((h * l) ^ -0.5);
	t_1 = (M * D) ^ 2.0;
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = abs(t_0);
	elseif (d <= 3e-271)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (t_1 / d));
	elseif (d <= 1.4e-259)
		tmp = t_0;
	elseif (d <= 5.4e+55)
		tmp = -0.125 * (t_1 * ((sqrt(h) / d) / (l ^ 1.5)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -5e-310], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, 3e-271], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-259], t$95$0, If[LessEqual[d, 5.4e+55], N[(-0.125 * N[(t$95$1 * N[(N[(N[Sqrt[h], $MachinePrecision] / d), $MachinePrecision] / N[Power[l, 1.5], $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 5 regimes

2. if d < -4.999999999999985e-310

   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 64.2%

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

   4. Taylor expanded in d around inf 8.7%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

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

      4. inv-pow 31.7%

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

      5. sqrt-pow1 31.7%

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

      6. metadata-eval 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. unpow2 31.7%

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

      2. rem-sqrt-square 45.4%

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

   8. Simplified 45.4%

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

   if -4.999999999999985e-310 < d < 3.00000000000000002e-271

   1. Initial program 25.0%

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

   3. Simplified 25.0%

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

      1. fma-udef 25.0%

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

      2. *-commutative 25.0%

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

      3. *-un-lft-identity 25.0%

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

      4. times-frac 25.0%

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

      5. metadata-eval 25.0%

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

      6. *-commutative 25.0%

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

   5. Applied egg-rr 25.0%

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

   6. Taylor expanded in d around 0 50.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. swap-sqr 50.8%

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

      5. unpow2 50.8%

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

      6. associate-/l* 50.8%

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

   8. Simplified 50.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 50.8%

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

      2. *-commutative 50.8%

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

      3. sqrt-div 50.8%

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

      4. sqrt-pow1 87.5%

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

      5. metadata-eval 87.5%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 87.5%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 3.00000000000000002e-271 < d < 1.4e-259

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

      1. fma-udef 18.3%

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

      2. *-commutative 18.3%

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

      3. *-un-lft-identity 18.3%

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

      4. times-frac 18.3%

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

      5. metadata-eval 18.3%

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

      6. *-commutative 18.3%

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

   5. Applied egg-rr 18.3%

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

   6. Taylor expanded in d around inf 60.3%

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

      1. unpow1/2 60.3%

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

      2. rem-exp-log 57.7%

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

      3. exp-neg 57.7%

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

      4. exp-prod 57.7%

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

      5. distribute-lft-neg-out 57.7%

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

      6. distribute-rgt-neg-in 57.7%

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

      7. metadata-eval 57.7%

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

      8. exp-to-pow 60.3%

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

   8. Simplified 60.3%

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

   if 1.4e-259 < d < 5.39999999999999954e55

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

      1. fma-udef 55.1%

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

      2. *-commutative 55.1%

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

      3. *-un-lft-identity 55.1%

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

      4. times-frac 55.1%

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

      5. metadata-eval 55.1%

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

      6. *-commutative 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in d around 0 32.7%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 32.5%

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

      2. unpow2 32.5%

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

      3. unpow2 32.5%

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

      4. swap-sqr 40.8%

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

      5. unpow2 40.8%

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

      6. associate-/l* 41.1%

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

   8. Simplified 41.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 40.7%

         \[\leadsto -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\right)\right)} \]

      2. expm1-udef 40.5%

         \[\leadsto -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\right)} - 1\right)} \]

      3. associate-/r/ 40.4%

         \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}\right)} - 1\right) \]

      4. *-commutative 40.4%

         \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left(M \cdot D\right)}}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} - 1\right) \]

      5. sqrt-div 41.9%

         \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right)} - 1\right) \]

      6. sqrt-pow1 54.2%

         \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right)} - 1\right) \]

      7. metadata-eval 54.2%

         \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right)} - 1\right) \]

   10. Applied egg-rr 54.2%

       \[\leadsto -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 58.5%

          \[\leadsto -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\right)} \]

       2. expm1-log1p 59.6%

          \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

       3. associate-/r/ 62.3%

          \[\leadsto -0.125 \cdot \color{blue}{\frac{{\left(M \cdot D\right)}^{2}}{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}} \]

       4. *-rgt-identity 62.3%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{{\left(M \cdot D\right)}^{2} \cdot 1}}{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}} \]

       5. associate-*r/ 62.4%

          \[\leadsto -0.125 \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot \frac{1}{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}\right)} \]

       6. associate-/r/ 62.3%

          \[\leadsto -0.125 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \color{blue}{\left(\frac{1}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}\right) \]

       7. *-commutative 62.3%

          \[\leadsto -0.125 \cdot \left({\color{blue}{\left(D \cdot M\right)}}^{2} \cdot \left(\frac{1}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\right) \]

       8. times-frac 63.5%

          \[\leadsto -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\frac{1 \cdot \sqrt{h}}{d \cdot {\ell}^{1.5}}}\right) \]

       9. *-lft-identity 63.5%

          \[\leadsto -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\color{blue}{\sqrt{h}}}{d \cdot {\ell}^{1.5}}\right) \]

       10. associate-/r* 62.5%

           \[\leadsto -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\frac{\frac{\sqrt{h}}{d}}{{\ell}^{1.5}}}\right) \]

   12. Simplified 62.5%

       \[\leadsto -0.125 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{\sqrt{h}}{d}}{{\ell}^{1.5}}\right)} \]

   if 5.39999999999999954e55 < d

   1. Initial program 72.2%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in d around inf 74.3%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 85.1%

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

   6. Applied egg-rr 85.1%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-271}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;-0.125 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \frac{\frac{\sqrt{h}}{d}}{{\ell}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (fabs (* d (pow (* h l) -0.5)))
   (if (<= d 5.4e+55)
     (* -0.125 (* (/ (sqrt h) (pow l 1.5)) (/ (pow (* M D) 2.0) d)))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = fabs((d * pow((h * l), -0.5)));
	} else if (d <= 5.4e+55) {
		tmp = -0.125 * ((sqrt(h) / pow(l, 1.5)) * (pow((M * D), 2.0) / d));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-5d-310)) then
        tmp = abs((d * ((h * l) ** (-0.5d0))))
    else if (d <= 5.4d+55) then
        tmp = (-0.125d0) * ((sqrt(h) / (l ** 1.5d0)) * (((m * d_1) ** 2.0d0) / d))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = Math.abs((d * Math.pow((h * l), -0.5)));
	} else if (d <= 5.4e+55) {
		tmp = -0.125 * ((Math.sqrt(h) / Math.pow(l, 1.5)) * (Math.pow((M * D), 2.0) / d));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = math.fabs((d * math.pow((h * l), -0.5)))
	elif d <= 5.4e+55:
		tmp = -0.125 * ((math.sqrt(h) / math.pow(l, 1.5)) * (math.pow((M * D), 2.0) / d))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = abs(Float64(d * (Float64(h * l) ^ -0.5)));
	elseif (d <= 5.4e+55)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64((Float64(M * D) ^ 2.0) / d)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = abs((d * ((h * l) ^ -0.5)));
	elseif (d <= 5.4e+55)
		tmp = -0.125 * ((sqrt(h) / (l ^ 1.5)) * (((M * D) ^ 2.0) / d));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, 5.4e+55], N[(-0.125 * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $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 < -4.999999999999985e-310

   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 64.2%

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

   4. Taylor expanded in d around inf 8.7%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

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

      4. inv-pow 31.7%

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

      5. sqrt-pow1 31.7%

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

      6. metadata-eval 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. unpow2 31.7%

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

      2. rem-sqrt-square 45.4%

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

   8. Simplified 45.4%

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

   if -4.999999999999985e-310 < d < 5.39999999999999954e55

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

      1. fma-udef 49.8%

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

      2. *-commutative 49.8%

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

      3. *-un-lft-identity 49.8%

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

      4. times-frac 49.8%

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

      5. metadata-eval 49.8%

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

      6. *-commutative 49.8%

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

   5. Applied egg-rr 49.8%

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

   6. Taylor expanded in d around 0 32.7%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 32.5%

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

      2. unpow2 32.5%

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

      3. unpow2 32.5%

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

      4. swap-sqr 39.5%

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

      5. unpow2 39.5%

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

      6. associate-/l* 39.8%

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

   8. Simplified 39.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. sqrt-div 41.9%

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

      4. sqrt-pow1 59.0%

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

      5. metadata-eval 59.0%

         \[\leadsto -0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   10. Applied egg-rr 59.0%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]

   if 5.39999999999999954e55 < d

   1. Initial program 72.2%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in d around inf 74.3%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 85.1%

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

   6. Applied egg-rr 85.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15: 46.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{-159}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.65e-159)
   (fabs (* d (pow (* h l) -0.5)))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.65e-159) {
		tmp = fabs((d * pow((h * l), -0.5)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 1.65d-159) then
        tmp = abs((d * ((h * l) ** (-0.5d0))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.65e-159) {
		tmp = Math.abs((d * Math.pow((h * l), -0.5)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.65e-159:
		tmp = math.fabs((d * math.pow((h * l), -0.5)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.65e-159)
		tmp = abs(Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.65e-159)
		tmp = abs((d * ((h * l) ^ -0.5)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.65e-159], N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $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 < 1.6500000000000001e-159

   1. Initial program 54.8%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in d around inf 11.0%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

      4. inv-pow 26.6%

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

      5. sqrt-pow1 26.6%

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

      6. metadata-eval 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. unpow2 26.6%

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

      2. rem-sqrt-square 38.8%

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

   8. Simplified 38.8%

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

   if 1.6500000000000001e-159 < d

   1. Initial program 72.2%

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

   3. Simplified 73.2%

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

      1. fma-udef 73.2%

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

      2. *-commutative 73.2%

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

      3. *-un-lft-identity 73.2%

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

      4. times-frac 73.2%

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

      5. metadata-eval 73.2%

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

      6. *-commutative 73.2%

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

   5. Applied egg-rr 73.2%

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

   6. Taylor expanded in d around inf 52.4%

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

      1. unpow1/2 52.4%

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

      2. rem-exp-log 50.2%

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

      3. exp-neg 50.3%

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

      4. exp-prod 50.2%

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

      5. distribute-lft-neg-out 50.2%

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

      6. distribute-rgt-neg-in 50.2%

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

      7. metadata-eval 50.2%

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

      8. exp-to-pow 52.4%

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

   8. Simplified 52.4%

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

      1. *-commutative 52.4%

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

      2. unpow-prod-down 63.7%

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

   10. Applied egg-rr 63.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{-159}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 16: 42.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;M \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* h l) -0.5)))) (if (<= M 3e+77) (fabs t_0) t_0)))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((h * l), -0.5);
	double tmp;
	if (M <= 3e+77) {
		tmp = fabs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((h * l) ** (-0.5d0))
    if (m <= 3d+77) then
        tmp = abs(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((h * l), -0.5);
	double tmp;
	if (M <= 3e+77) {
		tmp = Math.abs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.pow((h * l), -0.5)
	tmp = 0
	if M <= 3e+77:
		tmp = math.fabs(t_0)
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(h * l) ^ -0.5))
	tmp = 0.0
	if (M <= 3e+77)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * ((h * l) ^ -0.5);
	tmp = 0.0;
	if (M <= 3e+77)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 3e+77], N[Abs[t$95$0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9999999999999998e77

   1. Initial program 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in d around inf 29.5%

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

      1. add-sqr-sqrt 27.5%

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

      2. sqrt-unprod 33.8%

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

      3. pow2 33.8%

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

      4. inv-pow 33.8%

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

      5. sqrt-pow1 33.8%

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

      6. metadata-eval 33.8%

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

   6. Applied egg-rr 33.8%

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

      1. unpow2 33.8%

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

      2. rem-sqrt-square 49.4%

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

   8. Simplified 49.4%

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

   if 2.9999999999999998e77 < M

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

      1. fma-udef 75.2%

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

      2. *-commutative 75.2%

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

      3. *-un-lft-identity 75.2%

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

      4. times-frac 75.2%

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

      5. metadata-eval 75.2%

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

      6. *-commutative 75.2%

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

   5. Applied egg-rr 75.2%

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

   6. Taylor expanded in d around inf 22.6%

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

      1. unpow1/2 22.6%

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

      2. rem-exp-log 22.3%

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

      3. exp-neg 22.3%

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

      4. exp-prod 22.3%

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

      5. distribute-lft-neg-out 22.3%

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

      6. distribute-rgt-neg-in 22.3%

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

      7. metadata-eval 22.3%

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

      8. exp-to-pow 22.6%

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

   8. Simplified 22.6%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 17: 27.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{+127}:\\ \;\;\;\;d \cdot \sqrt{h \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -4e+127) (* d (sqrt (* h l))) (* d (sqrt (/ (/ 1.0 l) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -4e+127) {
		tmp = d * sqrt((h * l));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= (-4d+127)) then
        tmp = d * sqrt((h * l))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -4e+127) {
		tmp = d * Math.sqrt((h * l));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -4e+127:
		tmp = d * math.sqrt((h * l))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -4e+127)
		tmp = Float64(d * sqrt(Float64(h * l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -4e+127)
		tmp = d * sqrt((h * l));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -4e+127], N[(d * N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -3.99999999999999982e127

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in d around inf 3.4%

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

      1. add-exp-log 3.4%

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

      2. log-rec 3.4%

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

   6. Applied egg-rr 3.4%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 16.5%

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

      3. sqr-neg 16.5%

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

      4. sqrt-unprod 16.0%

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

      5. add-sqr-sqrt 16.5%

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

      6. *-commutative 16.5%

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

      7. add-exp-log 16.5%

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

   8. Applied egg-rr 16.5%

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

   if -3.99999999999999982e127 < h

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in d around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.4%

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

   6. Simplified 32.4%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{+127}:\\ \;\;\;\;d \cdot \sqrt{h \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 18: 27.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.35 \cdot 10^{+127}:\\ \;\;\;\;d \cdot \sqrt{h \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.35e+127) (* d (sqrt (* h l))) (* d (pow (* h l) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.35e+127) {
		tmp = d * sqrt((h * l));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= (-1.35d+127)) then
        tmp = d * sqrt((h * l))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.35e+127) {
		tmp = d * Math.sqrt((h * l));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.35e+127:
		tmp = d * math.sqrt((h * l))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.35e+127)
		tmp = Float64(d * sqrt(Float64(h * l)));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.35e+127)
		tmp = d * sqrt((h * l));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.35e+127], N[(d * N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.3500000000000001e127

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in d around inf 3.4%

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

      1. add-exp-log 3.4%

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

      2. log-rec 3.4%

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

   6. Applied egg-rr 3.4%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 16.5%

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

      3. sqr-neg 16.5%

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

      4. sqrt-unprod 16.0%

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

      5. add-sqr-sqrt 16.5%

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

      6. *-commutative 16.5%

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

      7. add-exp-log 16.5%

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

   8. Applied egg-rr 16.5%

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

   if -1.3500000000000001e127 < h

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. fma-udef 63.4%

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

      2. *-commutative 63.4%

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

      3. *-un-lft-identity 63.4%

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

      4. times-frac 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left({\left(D \cdot \color{blue}{\left(\frac{-0.5}{1} \cdot \frac{M}{d}\right)}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. metadata-eval 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left({\left(D \cdot \left(\color{blue}{-0.5} \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. *-commutative 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left({\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)} + 1\right) \]

   5. Applied egg-rr 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left({\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right) + 1\right)} \]

   6. Taylor expanded in d around inf 32.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow1/2 32.2%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

      2. rem-exp-log 31.1%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 31.1%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 31.1%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 31.1%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 31.1%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 31.1%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 32.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   8. Simplified 32.2%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.35 \cdot 10^{+127}:\\ \;\;\;\;d \cdot \sqrt{h \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19: 4.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{h \cdot \ell} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt((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 * sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

4. Taylor expanded in d around inf 28.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. add-exp-log 27.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right)}}} \]

   2. log-rec 27.2%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{-\log \left(h \cdot \ell\right)}}} \]

6. Applied egg-rr 27.2%

   \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 14.7%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{-\log \left(h \cdot \ell\right)} \cdot \sqrt{-\log \left(h \cdot \ell\right)}}}} \]

   2. sqrt-unprod 18.4%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{\left(-\log \left(h \cdot \ell\right)\right) \cdot \left(-\log \left(h \cdot \ell\right)\right)}}}} \]

   3. sqr-neg 18.4%

      \[\leadsto d \cdot \sqrt{e^{\sqrt{\color{blue}{\log \left(h \cdot \ell\right) \cdot \log \left(h \cdot \ell\right)}}}} \]

   4. sqrt-unprod 3.5%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{\sqrt{\log \left(h \cdot \ell\right)} \cdot \sqrt{\log \left(h \cdot \ell\right)}}}} \]

   5. add-sqr-sqrt 4.8%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{\log \left(h \cdot \ell\right)}}} \]

   6. *-commutative 4.8%

      \[\leadsto d \cdot \sqrt{e^{\log \color{blue}{\left(\ell \cdot h\right)}}} \]

   7. add-exp-log 4.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\ell \cdot h}} \]

8. Applied egg-rr 4.8%

   \[\leadsto d \cdot \sqrt{\color{blue}{\ell \cdot h}} \]

9. Final simplification 4.8%

   \[\leadsto d \cdot \sqrt{h \cdot \ell} \]

Reproduce

herbie shell --seed 2023333 
(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)))))