Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.3% → 78.1%

Time: 26.0s

Alternatives: 23

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: 65.3% 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: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (l <= (-1d-309)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (l <= -1e-309) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-309:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * 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))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 63.0%

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

      1. associate-*l* 62.3%

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

      2. metadata-eval 62.3%

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

      3. unpow1/2 62.3%

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

      4. metadata-eval 62.3%

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

      5. unpow1/2 62.3%

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

      6. associate-*l* 62.3%

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

      7. metadata-eval 62.3%

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

      8. times-frac 62.3%

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

   3. Simplified 62.3%

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

      1. frac-2neg 62.3%

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

      2. sqrt-div 78.4%

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

   5. Applied egg-rr 78.4%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.8%

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

      1. metadata-eval 67.8%

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

      2. unpow1/2 67.8%

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

      3. metadata-eval 67.8%

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

      4. unpow1/2 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-*l* 67.8%

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

      7. times-frac 67.9%

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

      8. metadata-eval 67.9%

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

   3. Simplified 67.9%

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

      1. expm1-log1p-u 41.9%

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

      2. expm1-udef 30.3%

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

   5. Applied egg-rr 36.8%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l/ 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 81.0%

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

Alternative 2: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.55 \cdot 10^{+165}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -3.55e+165)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -3.3e-165)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (pow (* (* (/ D d) (* 0.5 M)) (sqrt (/ h l))) 2.0)))))
     (if (<= d 8e-308)
       (-
        (* (sqrt (/ h (pow l 3.0))) (* (/ (* D D) (/ (/ d M) M)) 0.125))
        (* d (sqrt (/ 1.0 (* l h)))))
       (*
        (/ d (* (sqrt h) (sqrt l)))
        (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.55e+165) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -3.3e-165) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * pow((((D / d) * (0.5 * M)) * sqrt((h / l))), 2.0))));
	} else if (d <= 8e-308) {
		tmp = (sqrt((h / pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (d <= (-3.55d+165)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-3.3d-165)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((d_1 / d) * (0.5d0 * m)) * sqrt((h / l))) ** 2.0d0))))
    else if (d <= 8d-308) then
        tmp = (sqrt((h / (l ** 3.0d0))) * (((d_1 * d_1) / ((d / m) / m)) * 0.125d0)) - (d * sqrt((1.0d0 / (l * h))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (d <= -3.55e+165) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -3.3e-165) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * Math.pow((((D / d) * (0.5 * M)) * Math.sqrt((h / l))), 2.0))));
	} else if (d <= 8e-308) {
		tmp = (Math.sqrt((h / Math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * Math.sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -3.55e+165:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -3.3e-165:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * math.pow((((D / d) * (0.5 * M)) * math.sqrt((h / l))), 2.0))))
	elif d <= 8e-308:
		tmp = (math.sqrt((h / math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * math.sqrt((1.0 / (l * h))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.55e+165)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -3.3e-165)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (d <= 8e-308)
		tmp = Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(Float64(D * D) / Float64(Float64(d / M) / M)) * 0.125)) - Float64(d * sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -3.55e+165)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -3.3e-165)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((((D / d) * (0.5 * M)) * sqrt((h / l))) ^ 2.0))));
	elseif (d <= 8e-308)
		tmp = (sqrt((h / (l ^ 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -3.55e+165], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -3.3e-165], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-308], N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.54999999999999988e165

   1. Initial program 64.1%

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

      1. metadata-eval 64.1%

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

      2. unpow1/2 64.1%

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

      3. metadata-eval 64.1%

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

      4. unpow1/2 64.1%

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

      5. *-commutative 64.1%

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

      6. associate-*l* 64.1%

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

      7. times-frac 64.2%

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

      8. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

      1. clear-num 64.2%

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

      2. sqrt-div 64.1%

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

      3. metadata-eval 64.1%

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

   5. Applied egg-rr 64.1%

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

   6. Taylor expanded in d around -inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l/ 72.7%

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

   8. Simplified 72.7%

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

   if -3.54999999999999988e165 < d < -3.2999999999999998e-165

   1. Initial program 80.7%

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

      1. associate-*l* 80.7%

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

      2. metadata-eval 80.7%

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

      3. unpow1/2 80.7%

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

      4. metadata-eval 80.7%

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

      5. unpow1/2 80.7%

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

      6. associate-*l* 80.7%

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

      7. metadata-eval 80.7%

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

      8. times-frac 80.7%

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

   3. Simplified 80.7%

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

      1. add-sqr-sqrt 80.7%

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

      2. pow2 80.7%

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

      3. sqrt-prod 80.7%

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

      4. sqrt-pow1 82.9%

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

      5. metadata-eval 82.9%

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

      6. pow1 82.9%

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

      7. div-inv 82.9%

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

      8. metadata-eval 82.9%

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

   5. Applied egg-rr 82.9%

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

   if -3.2999999999999998e-165 < d < 8.00000000000000026e-308

   1. Initial program 30.3%

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

      1. metadata-eval 30.3%

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

      2. unpow1/2 30.3%

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

      3. metadata-eval 30.3%

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

      4. unpow1/2 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. times-frac 30.4%

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

      8. metadata-eval 30.4%

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

   3. Simplified 30.4%

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

      1. clear-num 30.4%

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

      2. sqrt-div 30.4%

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

      3. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

   6. Taylor expanded in d around -inf 47.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. *-commutative 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(-d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \]

      3. unsub-neg 47.3%

         \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot 0.125 - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. associate-*l* 47.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. associate-/l* 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. associate-/r* 53.9%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 8.00000000000000026e-308 < d

   1. Initial program 68.7%

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

      1. metadata-eval 68.7%

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

      2. unpow1/2 68.7%

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

      3. metadata-eval 68.7%

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

      4. unpow1/2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*l* 68.7%

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

      7. times-frac 68.8%

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

      8. metadata-eval 68.8%

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

   3. Simplified 68.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 30.7%

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

   5. Applied egg-rr 37.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 84.3%

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

      3. associate-/l* 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l/ 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.55 \cdot 10^{+165}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.65e-163)
   (*
    (sqrt (/ d h))
    (*
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))
     (/ (sqrt (- d)) (sqrt (- l)))))
   (if (<= d 8e-308)
     (-
      (* (sqrt (/ h (pow l 3.0))) (* (/ (* D D) (/ (/ d M) M)) 0.125))
      (* d (sqrt (/ 1.0 (* l h)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.65e-163) {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	} else if (d <= 8e-308) {
		tmp = (sqrt((h / pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (d <= (-2.65d-163)) then
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * (sqrt(-d) / sqrt(-l)))
    else if (d <= 8d-308) then
        tmp = (sqrt((h / (l ** 3.0d0))) * (((d_1 * d_1) / ((d / m) / m)) * 0.125d0)) - (d * sqrt((1.0d0 / (l * h))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (d <= -2.65e-163) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (d <= 8e-308) {
		tmp = (Math.sqrt((h / Math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * Math.sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.65e-163:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (math.sqrt(-d) / math.sqrt(-l)))
	elif d <= 8e-308:
		tmp = (math.sqrt((h / math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * math.sqrt((1.0 / (l * h))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.65e-163)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (d <= 8e-308)
		tmp = Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(Float64(D * D) / Float64(Float64(d / M) / M)) * 0.125)) - Float64(d * sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.65e-163)
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	elseif (d <= 8e-308)
		tmp = (sqrt((h / (l ^ 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.65e-163], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(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] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-308], N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.65000000000000008e-163

   1. Initial program 73.0%

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

      1. associate-*l* 72.0%

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

      2. metadata-eval 72.0%

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

      3. unpow1/2 72.0%

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

      4. metadata-eval 72.0%

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

      5. unpow1/2 72.0%

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

      6. associate-*l* 72.0%

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

      7. metadata-eval 72.0%

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

      8. times-frac 72.0%

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

   3. Simplified 72.0%

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

      1. frac-2neg 72.0%

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

      2. sqrt-div 74.9%

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

   5. Applied egg-rr 74.9%

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

   if -2.65000000000000008e-163 < d < 8.00000000000000026e-308

   1. Initial program 30.3%

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

      1. metadata-eval 30.3%

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

      2. unpow1/2 30.3%

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

      3. metadata-eval 30.3%

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

      4. unpow1/2 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. times-frac 30.4%

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

      8. metadata-eval 30.4%

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

   3. Simplified 30.4%

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

      1. clear-num 30.4%

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

      2. sqrt-div 30.4%

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

      3. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

   6. Taylor expanded in d around -inf 47.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. *-commutative 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(-d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \]

      3. unsub-neg 47.3%

         \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot 0.125 - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. associate-*l* 47.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. associate-/l* 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. associate-/r* 53.9%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 8.00000000000000026e-308 < d

   1. Initial program 68.7%

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

      1. metadata-eval 68.7%

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

      2. unpow1/2 68.7%

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

      3. metadata-eval 68.7%

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

      4. unpow1/2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*l* 68.7%

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

      7. times-frac 68.8%

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

      8. metadata-eval 68.8%

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

   3. Simplified 68.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 30.7%

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

   5. Applied egg-rr 37.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 84.3%

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

      3. associate-/l* 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l/ 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 4: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-164}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.8e-164)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (pow (* (* (/ D d) (* 0.5 M)) (sqrt (/ (* h 0.5) l))) 2.0)))
   (if (<= d 8e-308)
     (-
      (* (sqrt (/ h (pow l 3.0))) (* (/ (* D D) (/ (/ d M) M)) 0.125))
      (* d (sqrt (/ 1.0 (* l h)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-164) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - pow((((D / d) * (0.5 * M)) * sqrt(((h * 0.5) / l))), 2.0));
	} else if (d <= 8e-308) {
		tmp = (sqrt((h / pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (d <= (-2.8d-164)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((d_1 / d) * (0.5d0 * m)) * sqrt(((h * 0.5d0) / l))) ** 2.0d0))
    else if (d <= 8d-308) then
        tmp = (sqrt((h / (l ** 3.0d0))) * (((d_1 * d_1) / ((d / m) / m)) * 0.125d0)) - (d * sqrt((1.0d0 / (l * h))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (d <= -2.8e-164) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - Math.pow((((D / d) * (0.5 * M)) * Math.sqrt(((h * 0.5) / l))), 2.0));
	} else if (d <= 8e-308) {
		tmp = (Math.sqrt((h / Math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * Math.sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.8e-164:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - math.pow((((D / d) * (0.5 * M)) * math.sqrt(((h * 0.5) / l))), 2.0))
	elif d <= 8e-308:
		tmp = (math.sqrt((h / math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * math.sqrt((1.0 / (l * h))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.8e-164)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - (Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * sqrt(Float64(Float64(h * 0.5) / l))) ^ 2.0)));
	elseif (d <= 8e-308)
		tmp = Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(Float64(D * D) / Float64(Float64(d / M) / M)) * 0.125)) - Float64(d * sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.8e-164)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((D / d) * (0.5 * M)) * sqrt(((h * 0.5) / l))) ^ 2.0));
	elseif (d <= 8e-308)
		tmp = (sqrt((h / (l ^ 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.8e-164], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(h * 0.5), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-308], N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.8000000000000001e-164

   1. Initial program 73.0%

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

      1. metadata-eval 73.0%

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

      2. unpow1/2 73.0%

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

      3. metadata-eval 73.0%

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

      4. unpow1/2 73.0%

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

      5. *-commutative 73.0%

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

      6. associate-*l* 73.0%

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

      7. times-frac 73.0%

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

      8. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

      1. add-sqr-sqrt 73.1%

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

      2. pow2 73.1%

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

      3. sqrt-prod 73.1%

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

      4. sqrt-pow1 74.2%

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

      5. metadata-eval 74.2%

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

      6. pow1 74.2%

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

      7. div-inv 74.2%

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

      8. metadata-eval 74.2%

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

      9. associate-*r/ 74.2%

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

   5. Applied egg-rr 74.2%

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

   if -2.8000000000000001e-164 < d < 8.00000000000000026e-308

   1. Initial program 30.3%

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

      1. metadata-eval 30.3%

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

      2. unpow1/2 30.3%

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

      3. metadata-eval 30.3%

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

      4. unpow1/2 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. times-frac 30.4%

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

      8. metadata-eval 30.4%

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

   3. Simplified 30.4%

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

      1. clear-num 30.4%

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

      2. sqrt-div 30.4%

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

      3. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

   6. Taylor expanded in d around -inf 47.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. *-commutative 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(-d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \]

      3. unsub-neg 47.3%

         \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot 0.125 - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. associate-*l* 47.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. associate-/l* 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. associate-/r* 53.9%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 8.00000000000000026e-308 < d

   1. Initial program 68.7%

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

      1. metadata-eval 68.7%

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

      2. unpow1/2 68.7%

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

      3. metadata-eval 68.7%

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

      4. unpow1/2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*l* 68.7%

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

      7. times-frac 68.8%

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

      8. metadata-eval 68.8%

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

   3. Simplified 68.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 30.7%

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

   5. Applied egg-rr 37.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 84.3%

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

      3. associate-/l* 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l/ 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-164}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 5: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.5e-164)
   (*
    (/ (sqrt (/ d h)) (sqrt (/ l d)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* 0.5 (* M (/ D d))) 2.0)))))
   (if (<= d 8e-308)
     (-
      (* (sqrt (/ h (pow l 3.0))) (* (/ (* D D) (/ (/ d M) M)) 0.125))
      (* d (sqrt (/ 1.0 (* l h)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.5e-164) {
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 8e-308) {
		tmp = (sqrt((h / pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (d <= (-2.5d-164)) then
        tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0d0 - ((h / l) * (0.5d0 * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    else if (d <= 8d-308) then
        tmp = (sqrt((h / (l ** 3.0d0))) * (((d_1 * d_1) / ((d / m) / m)) * 0.125d0)) - (d * sqrt((1.0d0 / (l * h))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (d <= -2.5e-164) {
		tmp = (Math.sqrt((d / h)) / Math.sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * Math.pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 8e-308) {
		tmp = (Math.sqrt((h / Math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * Math.sqrt((1.0 / (l * h))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.5e-164:
		tmp = (math.sqrt((d / h)) / math.sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * math.pow((0.5 * (M * (D / d))), 2.0))))
	elif d <= 8e-308:
		tmp = (math.sqrt((h / math.pow(l, 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * math.sqrt((1.0 / (l * h))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.5e-164)
		tmp = Float64(Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	elseif (d <= 8e-308)
		tmp = Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(Float64(D * D) / Float64(Float64(d / M) / M)) * 0.125)) - Float64(d * sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.5e-164)
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * ((0.5 * (M * (D / d))) ^ 2.0))));
	elseif (d <= 8e-308)
		tmp = (sqrt((h / (l ^ 3.0))) * (((D * D) / ((d / M) / M)) * 0.125)) - (d * sqrt((1.0 / (l * h))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.5e-164], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-308], N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.49999999999999981e-164

   1. Initial program 73.0%

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

      1. metadata-eval 73.0%

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

      2. unpow1/2 73.0%

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

      3. metadata-eval 73.0%

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

      4. unpow1/2 73.0%

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

      5. *-commutative 73.0%

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

      6. associate-*l* 73.0%

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

      7. times-frac 73.0%

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

      8. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

      1. clear-num 73.0%

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

      2. sqrt-div 73.0%

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

      3. metadata-eval 73.0%

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

   5. Applied egg-rr 73.0%

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

      1. pow1 73.0%

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

      2. un-div-inv 73.1%

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

      3. div-inv 73.1%

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

      4. metadata-eval 73.1%

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

      5. associate-*l* 73.1%

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

   7. Applied egg-rr 73.1%

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

      1. unpow1 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. associate-*l* 73.1%

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

      5. *-commutative 73.1%

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

      6. associate-*l* 73.1%

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

   9. Simplified 73.1%

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

   if -2.49999999999999981e-164 < d < 8.00000000000000026e-308

   1. Initial program 30.3%

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

      1. metadata-eval 30.3%

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

      2. unpow1/2 30.3%

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

      3. metadata-eval 30.3%

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

      4. unpow1/2 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. times-frac 30.4%

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

      8. metadata-eval 30.4%

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

   3. Simplified 30.4%

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

      1. clear-num 30.4%

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

      2. sqrt-div 30.4%

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

      3. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

   6. Taylor expanded in d around -inf 47.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. *-commutative 47.3%

         \[\leadsto 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(-d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \]

      3. unsub-neg 47.3%

         \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      4. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot 0.125 - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. associate-*l* 47.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. associate-/l* 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. unpow2 47.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. associate-/r* 53.9%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 8.00000000000000026e-308 < d

   1. Initial program 68.7%

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

      1. metadata-eval 68.7%

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

      2. unpow1/2 68.7%

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

      3. metadata-eval 68.7%

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

      4. unpow1/2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*l* 68.7%

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

      7. times-frac 68.8%

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

      8. metadata-eval 68.8%

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

   3. Simplified 68.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 30.7%

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

   5. Applied egg-rr 37.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 84.3%

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

      3. associate-/l* 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l/ 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{\frac{\frac{d}{M}}{M}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 6: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.5e+161)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -6.6e-301)
     (*
      (* (sqrt (/ d h)) (/ 1.0 (sqrt (/ l d))))
      (- 1.0 (* (/ 0.125 l) (* (* (/ (* D D) d) (* M M)) (/ h d)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (pow (* D (* 0.5 (/ M d))) 2.0) (* h (/ 0.5 l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e+161) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow((D * (0.5 * (M / d))), 2.0) * (h * (0.5 / 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 <= (-6.5d+161)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-6.6d-301)) then
        tmp = (sqrt((d / h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - ((0.125d0 / l) * ((((d_1 * d_1) / d) * (m * m)) * (h / d))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (((d_1 * (0.5d0 * (m / d))) ** 2.0d0) * (h * (0.5d0 / 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 <= -6.5e+161) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (Math.sqrt((d / h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow((D * (0.5 * (M / d))), 2.0) * (h * (0.5 / l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -6.5e+161:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -6.6e-301:
		tmp = (math.sqrt((d / h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow((D * (0.5 * (M / d))), 2.0) * (h * (0.5 / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.5e+161)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -6.6e-301)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(Float64(0.125 / l) * Float64(Float64(Float64(Float64(D * D) / d) * Float64(M * M)) * Float64(h / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(h * Float64(0.5 / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -6.5e+161)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -6.6e-301)
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (((D * (0.5 * (M / d))) ^ 2.0) * (h * (0.5 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.5e+161], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -6.6e-301], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 / l), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.5e161

   1. Initial program 63.6%

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

      1. metadata-eval 63.6%

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

      2. unpow1/2 63.6%

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

      3. metadata-eval 63.6%

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

      4. unpow1/2 63.6%

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

      5. *-commutative 63.6%

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

      6. associate-*l* 63.6%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

      1. clear-num 63.8%

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

      2. sqrt-div 63.7%

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

      3. metadata-eval 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in d around -inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-/l/ 71.6%

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

   8. Simplified 71.6%

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

   if -6.5e161 < d < -6.6000000000000001e-301

   1. Initial program 66.3%

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

      1. metadata-eval 66.3%

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

      2. unpow1/2 66.3%

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

      3. metadata-eval 66.3%

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

      4. unpow1/2 66.3%

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

      5. *-commutative 66.3%

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

      6. associate-*l* 66.3%

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

      7. times-frac 66.4%

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

      8. metadata-eval 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.4%

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

      2. sqrt-div 66.3%

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

      3. metadata-eval 66.3%

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

   5. Applied egg-rr 66.3%

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

   6. Taylor expanded in M around 0 48.7%

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

      1. associate-*r/ 48.7%

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

      2. times-frac 50.1%

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

      3. associate-*r* 50.1%

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

      4. unpow2 50.1%

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

      5. times-frac 60.3%

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

      6. associate-/l* 59.0%

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

      7. associate-/r/ 60.3%

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

      8. unpow2 60.3%

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

      9. unpow2 60.3%

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

   8. Simplified 60.3%

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

   if -6.6000000000000001e-301 < d

   1. Initial program 65.9%

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

      1. metadata-eval 65.9%

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

      2. unpow1/2 65.9%

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

      3. metadata-eval 65.9%

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

      4. unpow1/2 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*l* 65.9%

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

      7. times-frac 66.0%

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

      8. metadata-eval 66.0%

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

   3. Simplified 66.0%

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

      1. expm1-log1p-u 40.7%

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

      2. expm1-udef 29.5%

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

   5. Applied egg-rr 35.8%

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

      1. expm1-def 50.2%

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

      2. expm1-log1p 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*r* 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*r/ 80.8%

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

      7. metadata-eval 80.8%

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

      8. times-frac 80.8%

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

      9. *-rgt-identity 80.8%

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

      10. associate-*l/ 80.2%

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

      11. *-commutative 80.2%

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

      12. *-lft-identity 80.2%

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

      13. *-commutative 80.2%

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

      14. times-frac 80.2%

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

      15. metadata-eval 80.2%

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

      16. associate-/l* 80.2%

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

      17. associate-/r/ 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \end{array} \]

Alternative 7: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.25 \cdot 10^{+161}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -3.25e+161)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -6.6e-301)
     (*
      (* (sqrt (/ d h)) (/ 1.0 (sqrt (/ l d))))
      (- 1.0 (* (/ 0.125 l) (* (* (/ (* D D) d) (* M M)) (/ h d)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.25e+161) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (d <= (-3.25d+161)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-6.6d-301)) then
        tmp = (sqrt((d / h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - ((0.125d0 / l) * ((((d_1 * d_1) / d) * (m * m)) * (h / d))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (d <= -3.25e+161) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (Math.sqrt((d / h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -3.25e+161:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -6.6e-301:
		tmp = (math.sqrt((d / h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.25e+161)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -6.6e-301)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(Float64(0.125 / l) * Float64(Float64(Float64(Float64(D * D) / d) * Float64(M * M)) * Float64(h / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -3.25e+161)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -6.6e-301)
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -3.25e+161], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -6.6e-301], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 / l), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.25e161

   1. Initial program 63.6%

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

      1. metadata-eval 63.6%

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

      2. unpow1/2 63.6%

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

      3. metadata-eval 63.6%

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

      4. unpow1/2 63.6%

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

      5. *-commutative 63.6%

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

      6. associate-*l* 63.6%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

      1. clear-num 63.8%

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

      2. sqrt-div 63.7%

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

      3. metadata-eval 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in d around -inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-/l/ 71.6%

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

   8. Simplified 71.6%

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

   if -3.25e161 < d < -6.6000000000000001e-301

   1. Initial program 66.3%

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

      1. metadata-eval 66.3%

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

      2. unpow1/2 66.3%

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

      3. metadata-eval 66.3%

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

      4. unpow1/2 66.3%

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

      5. *-commutative 66.3%

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

      6. associate-*l* 66.3%

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

      7. times-frac 66.4%

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

      8. metadata-eval 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.4%

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

      2. sqrt-div 66.3%

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

      3. metadata-eval 66.3%

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

   5. Applied egg-rr 66.3%

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

   6. Taylor expanded in M around 0 48.7%

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

      1. associate-*r/ 48.7%

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

      2. times-frac 50.1%

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

      3. associate-*r* 50.1%

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

      4. unpow2 50.1%

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

      5. times-frac 60.3%

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

      6. associate-/l* 59.0%

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

      7. associate-/r/ 60.3%

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

      8. unpow2 60.3%

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

      9. unpow2 60.3%

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

   8. Simplified 60.3%

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

   if -6.6000000000000001e-301 < d

   1. Initial program 65.9%

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

      1. metadata-eval 65.9%

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

      2. unpow1/2 65.9%

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

      3. metadata-eval 65.9%

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

      4. unpow1/2 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*l* 65.9%

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

      7. times-frac 66.0%

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

      8. metadata-eval 66.0%

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

   3. Simplified 66.0%

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

      1. expm1-log1p-u 40.7%

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

      2. expm1-udef 29.5%

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

   5. Applied egg-rr 35.8%

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

      1. expm1-def 50.2%

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

      2. expm1-log1p 80.9%

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

      3. associate-/l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-*l/ 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.25 \cdot 10^{+161}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 8: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2e-311)
   (*
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
    (sqrt (/ d h)))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2e-311) {
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * sqrt((d / h));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (l <= 2d-311) then
        tmp = (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))))) * sqrt((d / h))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (l <= 2e-311) {
		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 {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2e-311:
		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:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2e-311)
		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)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2e-311)
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))))) * sqrt((d / h));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2e-311], 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], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9999999999999e-311

   1. Initial program 63.0%

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

      1. associate-*l* 62.3%

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

      2. metadata-eval 62.3%

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

      3. unpow1/2 62.3%

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

      4. metadata-eval 62.3%

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

      5. unpow1/2 62.3%

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

      6. associate-*l* 62.3%

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

      7. metadata-eval 62.3%

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

      8. times-frac 62.3%

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

   3. Simplified 62.3%

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

   if 1.9999999999999e-311 < l

   1. Initial program 67.8%

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

      1. metadata-eval 67.8%

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

      2. unpow1/2 67.8%

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

      3. metadata-eval 67.8%

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

      4. unpow1/2 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-*l* 67.8%

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

      7. times-frac 67.9%

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

      8. metadata-eval 67.9%

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

   3. Simplified 67.9%

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

      1. expm1-log1p-u 41.9%

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

      2. expm1-udef 30.3%

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

   5. Applied egg-rr 36.8%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l/ 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 73.8%

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

Alternative 9: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* 0.5 (/ h l)))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (l <= (-1d-309)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (0.5d0 * (h / l))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (l <= -1e-309) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (Math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-309:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / l))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / 2.0) * (D / d)) ^ 2.0) * (0.5 * (h / l))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 63.0%

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

      1. metadata-eval 63.0%

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

      2. unpow1/2 63.0%

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

      3. metadata-eval 63.0%

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

      4. unpow1/2 63.0%

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

      5. *-commutative 63.0%

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

      6. associate-*l* 63.0%

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

      7. times-frac 63.1%

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

      8. metadata-eval 63.1%

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

   3. Simplified 63.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.8%

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

      1. metadata-eval 67.8%

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

      2. unpow1/2 67.8%

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

      3. metadata-eval 67.8%

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

      4. unpow1/2 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-*l* 67.8%

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

      7. times-frac 67.9%

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

      8. metadata-eval 67.9%

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

   3. Simplified 67.9%

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

      1. expm1-log1p-u 41.9%

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

      2. expm1-udef 30.3%

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

   5. Applied egg-rr 36.8%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l/ 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 74.2%

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

Alternative 10: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (/ (sqrt (/ d h)) (sqrt (/ l d)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* 0.5 (* M (/ D d))) 2.0)))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (pow (/ (* D (* 0.5 M)) d) 2.0) (/ 0.5 (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * pow((0.5 * (M * (D / d))), 2.0))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (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 (l <= (-1d-309)) then
        tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0d0 - ((h / l) * (0.5d0 * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (0.5d0 / (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 (l <= -1e-309) {
		tmp = (Math.sqrt((d / h)) / Math.sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * Math.pow((0.5 * (M * (D / d))), 2.0))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-309:
		tmp = (math.sqrt((d / h)) / math.sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * math.pow((0.5 * (M * (D / d))), 2.0))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow(((D * (0.5 * M)) / d), 2.0) * (0.5 / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - ((h / l) * (0.5 * ((0.5 * (M * (D / d))) ^ 2.0))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((((D * (0.5 * M)) / d) ^ 2.0) * (0.5 / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 63.0%

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

      1. metadata-eval 63.0%

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

      2. unpow1/2 63.0%

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

      3. metadata-eval 63.0%

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

      4. unpow1/2 63.0%

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

      5. *-commutative 63.0%

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

      6. associate-*l* 63.0%

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

      7. times-frac 63.1%

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

      8. metadata-eval 63.1%

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

   3. Simplified 63.1%

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

      1. clear-num 63.1%

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

      2. sqrt-div 63.1%

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

      3. metadata-eval 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. pow1 63.1%

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

      2. un-div-inv 63.1%

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

      3. div-inv 63.1%

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

      4. metadata-eval 63.1%

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

      5. associate-*l* 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. unpow1 63.1%

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

      2. *-commutative 63.1%

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

      3. *-commutative 63.1%

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

      4. associate-*l* 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*l* 63.1%

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

   9. Simplified 63.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.8%

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

      1. metadata-eval 67.8%

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

      2. unpow1/2 67.8%

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

      3. metadata-eval 67.8%

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

      4. unpow1/2 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-*l* 67.8%

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

      7. times-frac 67.9%

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

      8. metadata-eval 67.9%

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

   3. Simplified 67.9%

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

      1. expm1-log1p-u 41.9%

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

      2. expm1-udef 30.3%

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

   5. Applied egg-rr 36.8%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l/ 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 74.2%

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

Alternative 11: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{elif}\;d \leq 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* t_0 t_1)
          (- 1.0 (/ (* (* (* D D) 0.125) (* M (* h M))) (* l (* d d)))))))
   (if (<= d -2.7e+150)
     (* d (- (sqrt (/ (/ 1.0 h) l))))
     (if (<= d -9e-111)
       t_2
       (if (<= d -6.6e-301)
         (* t_1 (* t_0 (* -0.125 (/ (* D (* (/ h d) (/ M (/ d M)))) (/ l D)))))
         (if (<= d 1.26e-127)
           (* (/ (sqrt h) (pow l 1.5)) (/ (* (* D D) -0.125) (/ (/ d M) M)))
           (if (<= d 1e+85) t_2 (* d (* (pow h -0.5) (pow l -0.5))))))))))

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 = (t_0 * t_1) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	double tmp;
	if (d <= -2.7e+150) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -9e-111) {
		tmp = t_2;
	} else if (d <= -6.6e-301) {
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	} else if (d <= 1.26e-127) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	} else if (d <= 1e+85) {
		tmp = t_2;
	} else {
		tmp = d * (pow(h, -0.5) * pow(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 / l))
    t_1 = sqrt((d / h))
    t_2 = (t_0 * t_1) * (1.0d0 - ((((d_1 * d_1) * 0.125d0) * (m * (h * m))) / (l * (d * d))))
    if (d <= (-2.7d+150)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-9d-111)) then
        tmp = t_2
    else if (d <= (-6.6d-301)) then
        tmp = t_1 * (t_0 * ((-0.125d0) * ((d_1 * ((h / d) * (m / (d / m)))) / (l / d_1))))
    else if (d <= 1.26d-127) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * (((d_1 * d_1) * (-0.125d0)) / ((d / m) / m))
    else if (d <= 1d+85) then
        tmp = t_2
    else
        tmp = d * ((h ** (-0.5d0)) * (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 / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = (t_0 * t_1) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	double tmp;
	if (d <= -2.7e+150) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -9e-111) {
		tmp = t_2;
	} else if (d <= -6.6e-301) {
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	} else if (d <= 1.26e-127) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	} else if (d <= 1e+85) {
		tmp = t_2;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = (t_0 * t_1) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))))
	tmp = 0
	if d <= -2.7e+150:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -9e-111:
		tmp = t_2
	elif d <= -6.6e-301:
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))))
	elif d <= 1.26e-127:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M))
	elif d <= 1e+85:
		tmp = t_2
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(Float64(Float64(D * D) * 0.125) * Float64(M * Float64(h * M))) / Float64(l * Float64(d * d)))))
	tmp = 0.0
	if (d <= -2.7e+150)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -9e-111)
		tmp = t_2;
	elseif (d <= -6.6e-301)
		tmp = Float64(t_1 * Float64(t_0 * Float64(-0.125 * Float64(Float64(D * Float64(Float64(h / d) * Float64(M / Float64(d / M)))) / Float64(l / D)))));
	elseif (d <= 1.26e-127)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(Float64(Float64(D * D) * -0.125) / Float64(Float64(d / M) / M)));
	elseif (d <= 1e+85)
		tmp = t_2;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = (t_0 * t_1) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	tmp = 0.0;
	if (d <= -2.7e+150)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -9e-111)
		tmp = t_2;
	elseif (d <= -6.6e-301)
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	elseif (d <= 1.26e-127)
		tmp = (sqrt(h) / (l ^ 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	elseif (d <= 1e+85)
		tmp = t_2;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(D * D), $MachinePrecision] * 0.125), $MachinePrecision] * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.7e+150], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -9e-111], t$95$2, If[LessEqual[d, -6.6e-301], N[(t$95$1 * N[(t$95$0 * N[(-0.125 * N[(N[(D * N[(N[(h / d), $MachinePrecision] * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.26e-127], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e+85], t$95$2, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.70000000000000008e150

   1. Initial program 65.3%

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

      1. metadata-eval 65.3%

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

      2. unpow1/2 65.3%

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

      3. metadata-eval 65.3%

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

      4. unpow1/2 65.3%

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

      5. *-commutative 65.3%

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

      6. associate-*l* 65.3%

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

      7. times-frac 65.4%

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

      8. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

      1. clear-num 65.5%

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

      2. sqrt-div 65.3%

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

      3. metadata-eval 65.3%

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

   5. Applied egg-rr 65.3%

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

   6. Taylor expanded in d around -inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. associate-/l/ 68.4%

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

   8. Simplified 68.4%

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

   if -2.70000000000000008e150 < d < -8.99999999999999987e-111 or 1.2600000000000001e-127 < d < 1e85

   1. Initial program 79.3%

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

      1. metadata-eval 79.3%

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

      2. unpow1/2 79.3%

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

      3. metadata-eval 79.3%

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

      4. unpow1/2 79.3%

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

      5. *-commutative 79.3%

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

      6. associate-*l* 79.3%

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

      7. times-frac 79.4%

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

      8. metadata-eval 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. associate-*r* 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

      5. associate-*l* 75.3%

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

      6. unpow2 75.3%

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

   6. Simplified 75.3%

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

   if -8.99999999999999987e-111 < d < -6.6000000000000001e-301

   1. Initial program 48.8%

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

      1. associate-*l* 48.9%

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

      2. metadata-eval 48.9%

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

      3. unpow1/2 48.9%

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

      4. metadata-eval 48.9%

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

      5. unpow1/2 48.9%

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

      6. sub-neg 48.9%

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

      7. +-commutative 48.9%

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

      8. *-commutative 48.9%

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

      9. distribute-rgt-neg-in 48.9%

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

      10. fma-def 48.9%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in h around inf 28.8%

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

      1. *-commutative 28.8%

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

      2. times-frac 28.8%

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

      3. unpow2 28.8%

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

      4. associate-/l* 31.8%

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

      5. unpow2 31.8%

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

      6. times-frac 35.3%

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

      7. unpow2 35.3%

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

   6. Simplified 35.3%

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

      1. associate-*l/ 35.3%

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

      2. associate-/l* 40.8%

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

   8. Applied egg-rr 40.8%

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

   if -6.6000000000000001e-301 < d < 1.2600000000000001e-127

   1. Initial program 48.6%

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

      1. metadata-eval 48.6%

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

      2. unpow1/2 48.6%

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

      3. metadata-eval 48.6%

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

      4. unpow1/2 48.6%

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

      5. *-commutative 48.6%

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

      6. associate-*l* 48.6%

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

      7. times-frac 48.7%

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

      8. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

      1. clear-num 48.7%

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

      2. sqrt-div 48.8%

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

      3. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in d around 0 44.8%

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

      1. associate-*r* 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-/l* 44.7%

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

      4. associate-*r/ 44.7%

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

      5. unpow2 44.7%

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

      6. unpow2 44.7%

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

      7. associate-/r* 48.9%

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

   8. Simplified 48.9%

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

      1. sqrt-div 51.1%

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

   10. Applied egg-rr 51.1%

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

       1. sqr-pow 51.1%

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

       2. rem-sqrt-square 53.2%

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

       3. sqr-pow 53.2%

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

       4. fabs-sqr 53.2%

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

       5. sqr-pow 53.2%

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

       6. metadata-eval 53.2%

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

   12. Simplified 53.2%

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

   if 1e85 < d

   1. Initial program 72.1%

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

      1. metadata-eval 72.1%

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

      2. unpow1/2 72.1%

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

      3. metadata-eval 72.1%

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

      4. unpow1/2 72.1%

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

      5. *-commutative 72.1%

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

      6. associate-*l* 72.1%

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

      7. times-frac 72.1%

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

      8. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in d around inf 76.5%

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

      1. *-commutative 76.5%

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

      2. add-cbrt-cube 58.4%

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

      3. expm1-log1p-u 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 48.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 56.6%

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

      6. *-commutative 56.6%

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

      7. associate-/r* 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. expm1-def 76.2%

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

      2. expm1-log1p 77.4%

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

      3. associate-/l/ 76.5%

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

      4. unpow-1 76.5%

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

      5. sqr-pow 76.6%

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

      6. rem-sqrt-square 76.6%

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

      7. sqr-pow 76.5%

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

      8. fabs-sqr 76.5%

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

      9. sqr-pow 76.6%

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

      10. metadata-eval 76.6%

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

   8. Simplified 76.6%

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

      1. unpow-prod-down 81.6%

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

   10. Applied egg-rr 81.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{elif}\;d \leq 10^{+85}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_0\right) \cdot \left(1 - \frac{\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= d -1.45e+158)
     (* d (- (sqrt (/ (/ 1.0 h) l))))
     (if (<= d -6.6e-301)
       (*
        (* t_0 (/ 1.0 (sqrt (/ l d))))
        (- 1.0 (* (/ 0.125 l) (* (* (/ (* D D) d) (* M M)) (/ h d)))))
       (if (<= d 7.2e-128)
         (* (/ (sqrt h) (pow l 1.5)) (/ (* (* D D) -0.125) (/ (/ d M) M)))
         (if (<= d 2.3e+90)
           (*
            (* (sqrt (/ d l)) t_0)
            (- 1.0 (/ (* (* (* D D) 0.125) (* M (* h M))) (* l (* d d)))))
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (d <= -1.45e+158) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (t_0 * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else if (d <= 7.2e-128) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	} else if (d <= 2.3e+90) {
		tmp = (sqrt((d / l)) * t_0) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(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) :: tmp
    t_0 = sqrt((d / h))
    if (d <= (-1.45d+158)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-6.6d-301)) then
        tmp = (t_0 * (1.0d0 / sqrt((l / d)))) * (1.0d0 - ((0.125d0 / l) * ((((d_1 * d_1) / d) * (m * m)) * (h / d))))
    else if (d <= 7.2d-128) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * (((d_1 * d_1) * (-0.125d0)) / ((d / m) / m))
    else if (d <= 2.3d+90) then
        tmp = (sqrt((d / l)) * t_0) * (1.0d0 - ((((d_1 * d_1) * 0.125d0) * (m * (h * m))) / (l * (d * d))))
    else
        tmp = d * ((h ** (-0.5d0)) * (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 tmp;
	if (d <= -1.45e+158) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -6.6e-301) {
		tmp = (t_0 * (1.0 / Math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	} else if (d <= 7.2e-128) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	} else if (d <= 2.3e+90) {
		tmp = (Math.sqrt((d / l)) * t_0) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if d <= -1.45e+158:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -6.6e-301:
		tmp = (t_0 * (1.0 / math.sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))))
	elif d <= 7.2e-128:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (((D * D) * -0.125) / ((d / M) / M))
	elif d <= 2.3e+90:
		tmp = (math.sqrt((d / l)) * t_0) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -1.45e+158)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -6.6e-301)
		tmp = Float64(Float64(t_0 * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(Float64(0.125 / l) * Float64(Float64(Float64(Float64(D * D) / d) * Float64(M * M)) * Float64(h / d)))));
	elseif (d <= 7.2e-128)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(Float64(Float64(D * D) * -0.125) / Float64(Float64(d / M) / M)));
	elseif (d <= 2.3e+90)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(D * D) * 0.125) * Float64(M * Float64(h * M))) / Float64(l * Float64(d * d)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (d <= -1.45e+158)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -6.6e-301)
		tmp = (t_0 * (1.0 / sqrt((l / d)))) * (1.0 - ((0.125 / l) * ((((D * D) / d) * (M * M)) * (h / d))));
	elseif (d <= 7.2e-128)
		tmp = (sqrt(h) / (l ^ 1.5)) * (((D * D) * -0.125) / ((d / M) / M));
	elseif (d <= 2.3e+90)
		tmp = (sqrt((d / l)) * t_0) * (1.0 - ((((D * D) * 0.125) * (M * (h * M))) / (l * (d * d))));
	else
		tmp = d * ((h ^ -0.5) * (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]}, If[LessEqual[d, -1.45e+158], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -6.6e-301], N[(N[(t$95$0 * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 / l), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e-128], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e+90], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(D * D), $MachinePrecision] * 0.125), $MachinePrecision] * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.45000000000000012e158

   1. Initial program 63.6%

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

      1. metadata-eval 63.6%

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

      2. unpow1/2 63.6%

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

      3. metadata-eval 63.6%

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

      4. unpow1/2 63.6%

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

      5. *-commutative 63.6%

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

      6. associate-*l* 63.6%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

      1. clear-num 63.8%

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

      2. sqrt-div 63.7%

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

      3. metadata-eval 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in d around -inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-/l/ 71.6%

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

   8. Simplified 71.6%

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

   if -1.45000000000000012e158 < d < -6.6000000000000001e-301

   1. Initial program 66.3%

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

      1. metadata-eval 66.3%

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

      2. unpow1/2 66.3%

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

      3. metadata-eval 66.3%

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

      4. unpow1/2 66.3%

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

      5. *-commutative 66.3%

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

      6. associate-*l* 66.3%

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

      7. times-frac 66.4%

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

      8. metadata-eval 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.4%

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

      2. sqrt-div 66.3%

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

      3. metadata-eval 66.3%

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

   5. Applied egg-rr 66.3%

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

   6. Taylor expanded in M around 0 48.7%

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

      1. associate-*r/ 48.7%

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

      2. times-frac 50.1%

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

      3. associate-*r* 50.1%

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

      4. unpow2 50.1%

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

      5. times-frac 60.3%

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

      6. associate-/l* 59.0%

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

      7. associate-/r/ 60.3%

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

      8. unpow2 60.3%

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

      9. unpow2 60.3%

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

   8. Simplified 60.3%

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

   if -6.6000000000000001e-301 < d < 7.20000000000000049e-128

   1. Initial program 48.6%

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

      1. metadata-eval 48.6%

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

      2. unpow1/2 48.6%

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

      3. metadata-eval 48.6%

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

      4. unpow1/2 48.6%

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

      5. *-commutative 48.6%

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

      6. associate-*l* 48.6%

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

      7. times-frac 48.7%

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

      8. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

      1. clear-num 48.7%

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

      2. sqrt-div 48.8%

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

      3. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in d around 0 44.8%

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

      1. associate-*r* 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-/l* 44.7%

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

      4. associate-*r/ 44.7%

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

      5. unpow2 44.7%

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

      6. unpow2 44.7%

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

      7. associate-/r* 48.9%

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

   8. Simplified 48.9%

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

      1. sqrt-div 51.1%

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

   10. Applied egg-rr 51.1%

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

       1. sqr-pow 51.1%

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

       2. rem-sqrt-square 53.2%

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

       3. sqr-pow 53.2%

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

       4. fabs-sqr 53.2%

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

       5. sqr-pow 53.2%

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

       6. metadata-eval 53.2%

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

   12. Simplified 53.2%

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

   if 7.20000000000000049e-128 < d < 2.3e90

   1. Initial program 76.1%

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

      1. metadata-eval 76.1%

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

      2. unpow1/2 76.1%

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

      3. metadata-eval 76.1%

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

      4. unpow1/2 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*l* 76.1%

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

      7. times-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*r* 69.4%

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

      3. unpow2 69.4%

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

      4. unpow2 69.4%

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

      5. associate-*l* 69.4%

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

      6. unpow2 69.4%

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

   6. Simplified 69.4%

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

   if 2.3e90 < d

   1. Initial program 72.1%

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

      1. metadata-eval 72.1%

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

      2. unpow1/2 72.1%

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

      3. metadata-eval 72.1%

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

      4. unpow1/2 72.1%

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

      5. *-commutative 72.1%

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

      6. associate-*l* 72.1%

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

      7. times-frac 72.1%

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

      8. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in d around inf 76.5%

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

      1. *-commutative 76.5%

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

      2. add-cbrt-cube 58.4%

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

      3. expm1-log1p-u 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 48.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 56.6%

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

      6. *-commutative 56.6%

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

      7. associate-/r* 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. expm1-def 76.2%

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

      2. expm1-log1p 77.4%

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

      3. associate-/l/ 76.5%

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

      4. unpow-1 76.5%

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

      5. sqr-pow 76.6%

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

      6. rem-sqrt-square 76.6%

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

      7. sqr-pow 76.5%

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

      8. fabs-sqr 76.5%

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

      9. sqr-pow 76.6%

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

      10. metadata-eval 76.6%

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

   8. Simplified 76.6%

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

      1. unpow-prod-down 81.6%

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

   10. Applied egg-rr 81.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 13: 41.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;M \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \leq 4.6 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (sqrt (/ d h))) (t_2 (* t_0 t_1)))
   (if (<= M 2.9e-198)
     t_2
     (if (<= M 4.6e-167)
       (* d (* (pow h -0.5) (pow l -0.5)))
       (if (<= M 1.05e+15)
         t_2
         (*
          t_1
          (* t_0 (* -0.125 (* (/ D (/ l D)) (* (/ h d) (/ (* M M) d)))))))))))

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 = t_0 * t_1;
	double tmp;
	if (M <= 2.9e-198) {
		tmp = t_2;
	} else if (M <= 4.6e-167) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else if (M <= 1.05e+15) {
		tmp = t_2;
	} else {
		tmp = t_1 * (t_0 * (-0.125 * ((D / (l / D)) * ((h / d) * ((M * M) / d)))));
	}
	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 / l))
    t_1 = sqrt((d / h))
    t_2 = t_0 * t_1
    if (m <= 2.9d-198) then
        tmp = t_2
    else if (m <= 4.6d-167) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else if (m <= 1.05d+15) then
        tmp = t_2
    else
        tmp = t_1 * (t_0 * ((-0.125d0) * ((d_1 / (l / d_1)) * ((h / d) * ((m * m) / d)))))
    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 / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = t_0 * t_1;
	double tmp;
	if (M <= 2.9e-198) {
		tmp = t_2;
	} else if (M <= 4.6e-167) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else if (M <= 1.05e+15) {
		tmp = t_2;
	} else {
		tmp = t_1 * (t_0 * (-0.125 * ((D / (l / D)) * ((h / d) * ((M * M) / d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = t_0 * t_1
	tmp = 0
	if M <= 2.9e-198:
		tmp = t_2
	elif M <= 4.6e-167:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	elif M <= 1.05e+15:
		tmp = t_2
	else:
		tmp = t_1 * (t_0 * (-0.125 * ((D / (l / D)) * ((h / d) * ((M * M) / d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (M <= 2.9e-198)
		tmp = t_2;
	elseif (M <= 4.6e-167)
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	elseif (M <= 1.05e+15)
		tmp = t_2;
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(-0.125 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(h / d) * Float64(Float64(M * M) / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (M <= 2.9e-198)
		tmp = t_2;
	elseif (M <= 4.6e-167)
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	elseif (M <= 1.05e+15)
		tmp = t_2;
	else
		tmp = t_1 * (t_0 * (-0.125 * ((D / (l / D)) * ((h / d) * ((M * M) / d)))));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[M, 2.9e-198], t$95$2, If[LessEqual[M, 4.6e-167], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.05e+15], t$95$2, N[(t$95$1 * N[(t$95$0 * N[(-0.125 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 2.90000000000000001e-198 or 4.6000000000000003e-167 < M < 1.05e15

   1. Initial program 66.3%

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

      1. associate-*l* 65.9%

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

      2. metadata-eval 65.9%

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

      3. unpow1/2 65.9%

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

      4. metadata-eval 65.9%

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

      5. unpow1/2 65.9%

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

      6. associate-*l* 65.9%

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

      7. metadata-eval 65.9%

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

      8. times-frac 66.0%

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

   3. Simplified 66.0%

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

      1. sqrt-div 43.3%

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

   5. Applied egg-rr 43.3%

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

   6. Taylor expanded in M around 0 29.0%

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

      1. expm1-log1p-u 28.0%

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

      2. expm1-udef 21.3%

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

      3. sqrt-undiv 34.1%

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

   8. Applied egg-rr 34.1%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 46.5%

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

   10. Simplified 46.5%

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

   if 2.90000000000000001e-198 < M < 4.6000000000000003e-167

   1. Initial program 29.9%

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

      1. metadata-eval 29.9%

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

      2. unpow1/2 29.9%

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

      3. metadata-eval 29.9%

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

      4. unpow1/2 29.9%

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

      5. *-commutative 29.9%

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

      6. associate-*l* 29.9%

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

      7. times-frac 29.9%

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

      8. metadata-eval 29.9%

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

   3. Simplified 29.9%

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

   4. Taylor expanded in d around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. add-cbrt-cube 63.3%

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

      3. expm1-log1p-u 62.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 36.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 47.1%

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

      6. *-commutative 47.1%

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

      7. associate-/r* 47.1%

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

   6. Applied egg-rr 47.1%

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

      1. expm1-def 85.5%

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

      2. expm1-log1p 87.1%

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

      3. associate-/l/ 87.1%

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

      4. unpow-1 87.1%

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

      5. sqr-pow 87.3%

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

      6. rem-sqrt-square 87.3%

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

      7. sqr-pow 86.9%

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

      8. fabs-sqr 86.9%

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

      9. sqr-pow 87.3%

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

      10. metadata-eval 87.3%

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

   8. Simplified 87.3%

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

      1. unpow-prod-down 87.3%

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

   10. Applied egg-rr 87.3%

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

   if 1.05e15 < M

   1. Initial program 68.5%

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

      1. associate-*l* 68.5%

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

      2. metadata-eval 68.5%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 68.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 68.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 68.5%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 32.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 32.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. times-frac 32.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot -0.125\right)\right) \]

      3. unpow2 32.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.125\right)\right) \]

      4. associate-/l* 32.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.125\right)\right) \]

      5. unpow2 32.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h \cdot {M}^{2}}{\color{blue}{d \cdot d}}\right) \cdot -0.125\right)\right) \]

      6. times-frac 38.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{{M}^{2}}{d}\right)}\right) \cdot -0.125\right)\right) \]

      7. unpow2 38.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \cdot -0.125\right)\right) \]

   6. Simplified 38.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 4.6 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 14: 41.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;M \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)}{\frac{\ell}{D}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (sqrt (/ d h))) (t_2 (* t_0 t_1)))
   (if (<= M 2.9e-198)
     t_2
     (if (<= M 4.2e-167)
       (* d (* (pow h -0.5) (pow l -0.5)))
       (if (<= M 1.1e+19)
         t_2
         (*
          t_1
          (* t_0 (* -0.125 (/ (* D (* (/ h d) (/ M (/ d M)))) (/ l D))))))))))

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 = t_0 * t_1;
	double tmp;
	if (M <= 2.9e-198) {
		tmp = t_2;
	} else if (M <= 4.2e-167) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else if (M <= 1.1e+19) {
		tmp = t_2;
	} else {
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	}
	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 / l))
    t_1 = sqrt((d / h))
    t_2 = t_0 * t_1
    if (m <= 2.9d-198) then
        tmp = t_2
    else if (m <= 4.2d-167) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else if (m <= 1.1d+19) then
        tmp = t_2
    else
        tmp = t_1 * (t_0 * ((-0.125d0) * ((d_1 * ((h / d) * (m / (d / m)))) / (l / d_1))))
    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 / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = t_0 * t_1;
	double tmp;
	if (M <= 2.9e-198) {
		tmp = t_2;
	} else if (M <= 4.2e-167) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else if (M <= 1.1e+19) {
		tmp = t_2;
	} else {
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = t_0 * t_1
	tmp = 0
	if M <= 2.9e-198:
		tmp = t_2
	elif M <= 4.2e-167:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	elif M <= 1.1e+19:
		tmp = t_2
	else:
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (M <= 2.9e-198)
		tmp = t_2;
	elseif (M <= 4.2e-167)
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	elseif (M <= 1.1e+19)
		tmp = t_2;
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(-0.125 * Float64(Float64(D * Float64(Float64(h / d) * Float64(M / Float64(d / M)))) / Float64(l / D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (M <= 2.9e-198)
		tmp = t_2;
	elseif (M <= 4.2e-167)
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	elseif (M <= 1.1e+19)
		tmp = t_2;
	else
		tmp = t_1 * (t_0 * (-0.125 * ((D * ((h / d) * (M / (d / M)))) / (l / D))));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[M, 2.9e-198], t$95$2, If[LessEqual[M, 4.2e-167], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.1e+19], t$95$2, N[(t$95$1 * N[(t$95$0 * N[(-0.125 * N[(N[(D * N[(N[(h / d), $MachinePrecision] * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 2.90000000000000001e-198 or 4.20000000000000035e-167 < M < 1.1e19

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.7%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 65.7%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 65.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 65.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 65.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. associate-*l* 65.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      7. metadata-eval 65.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. times-frac 65.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sqrt-div 43.3%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in M around 0 29.2%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]

      2. expm1-udef 21.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}} \]

      3. sqrt-undiv 34.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h}}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. expm1-def 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]

      2. expm1-log1p 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Simplified 46.5%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   if 2.90000000000000001e-198 < M < 4.20000000000000035e-167

   1. Initial program 29.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 29.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 29.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 29.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 87.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 63.3%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 62.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 36.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 47.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 47.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 47.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 85.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 87.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 87.1%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 87.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 87.3%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 87.3%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 86.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 86.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 87.3%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 87.3%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 87.3%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 87.3%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   if 1.1e19 < M

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 69.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 69.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 69.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 69.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 69.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 33.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. times-frac 33.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot -0.125\right)\right) \]

      3. unpow2 33.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.125\right)\right) \]

      4. associate-/l* 34.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.125\right)\right) \]

      5. unpow2 34.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h \cdot {M}^{2}}{\color{blue}{d \cdot d}}\right) \cdot -0.125\right)\right) \]

      6. times-frac 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{{M}^{2}}{d}\right)}\right) \cdot -0.125\right)\right) \]

      7. unpow2 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \cdot -0.125\right)\right) \]

   6. Simplified 39.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 40.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{D \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)}{\frac{\ell}{D}}} \cdot -0.125\right)\right) \]

      2. associate-/l* 43.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{D \cdot \left(\frac{h}{d} \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)}{\frac{\ell}{D}} \cdot -0.125\right)\right) \]

   8. Applied egg-rr 43.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{D \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)}{\frac{\ell}{D}}} \cdot -0.125\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)}{\frac{\ell}{D}}\right)\right)\\ \end{array} \]

Alternative 15: 48.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(D \cdot D\right) \cdot -0.125\\ t_1 := \frac{\frac{1}{h}}{\ell}\\ t_2 := \frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{t_0}{\frac{\frac{d}{M}}{M}}\\ t_3 := d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{-72}:\\ \;\;\;\;d \cdot \left(-\sqrt{t_1}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({t_1}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{t_0}{\frac{d}{M}}\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* D D) -0.125))
        (t_1 (/ (/ 1.0 h) l))
        (t_2 (* (/ (sqrt h) (pow l 1.5)) (/ t_0 (/ (/ d M) M))))
        (t_3 (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))
   (if (<= d -5.2e-72)
     (* d (- (sqrt t_1)))
     (if (<= d -4e-310)
       (* d (pow (pow t_1 3.0) 0.16666666666666666))
       (if (<= d 2.55e-160)
         t_2
         (if (<= d 3.3e-84)
           t_3
           (if (<= d 3.1e-39)
             (* (sqrt (/ h (pow l 3.0))) (* M (/ t_0 (/ d M))))
             (if (<= d 6.8e+34)
               t_3
               (if (<= d 2.8e+80)
                 t_2
                 (* d (* (pow h -0.5) (pow l -0.5))))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * D) * -0.125;
	double t_1 = (1.0 / h) / l;
	double t_2 = (sqrt(h) / pow(l, 1.5)) * (t_0 / ((d / M) / M));
	double t_3 = d * (sqrt((1.0 / l)) / sqrt(h));
	double tmp;
	if (d <= -5.2e-72) {
		tmp = d * -sqrt(t_1);
	} else if (d <= -4e-310) {
		tmp = d * pow(pow(t_1, 3.0), 0.16666666666666666);
	} else if (d <= 2.55e-160) {
		tmp = t_2;
	} else if (d <= 3.3e-84) {
		tmp = t_3;
	} else if (d <= 3.1e-39) {
		tmp = sqrt((h / pow(l, 3.0))) * (M * (t_0 / (d / M)));
	} else if (d <= 6.8e+34) {
		tmp = t_3;
	} else if (d <= 2.8e+80) {
		tmp = t_2;
	} else {
		tmp = d * (pow(h, -0.5) * pow(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) :: t_3
    real(8) :: tmp
    t_0 = (d_1 * d_1) * (-0.125d0)
    t_1 = (1.0d0 / h) / l
    t_2 = (sqrt(h) / (l ** 1.5d0)) * (t_0 / ((d / m) / m))
    t_3 = d * (sqrt((1.0d0 / l)) / sqrt(h))
    if (d <= (-5.2d-72)) then
        tmp = d * -sqrt(t_1)
    else if (d <= (-4d-310)) then
        tmp = d * ((t_1 ** 3.0d0) ** 0.16666666666666666d0)
    else if (d <= 2.55d-160) then
        tmp = t_2
    else if (d <= 3.3d-84) then
        tmp = t_3
    else if (d <= 3.1d-39) then
        tmp = sqrt((h / (l ** 3.0d0))) * (m * (t_0 / (d / m)))
    else if (d <= 6.8d+34) then
        tmp = t_3
    else if (d <= 2.8d+80) then
        tmp = t_2
    else
        tmp = d * ((h ** (-0.5d0)) * (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 = (D * D) * -0.125;
	double t_1 = (1.0 / h) / l;
	double t_2 = (Math.sqrt(h) / Math.pow(l, 1.5)) * (t_0 / ((d / M) / M));
	double t_3 = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	double tmp;
	if (d <= -5.2e-72) {
		tmp = d * -Math.sqrt(t_1);
	} else if (d <= -4e-310) {
		tmp = d * Math.pow(Math.pow(t_1, 3.0), 0.16666666666666666);
	} else if (d <= 2.55e-160) {
		tmp = t_2;
	} else if (d <= 3.3e-84) {
		tmp = t_3;
	} else if (d <= 3.1e-39) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (M * (t_0 / (d / M)));
	} else if (d <= 6.8e+34) {
		tmp = t_3;
	} else if (d <= 2.8e+80) {
		tmp = t_2;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D * D) * -0.125
	t_1 = (1.0 / h) / l
	t_2 = (math.sqrt(h) / math.pow(l, 1.5)) * (t_0 / ((d / M) / M))
	t_3 = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	tmp = 0
	if d <= -5.2e-72:
		tmp = d * -math.sqrt(t_1)
	elif d <= -4e-310:
		tmp = d * math.pow(math.pow(t_1, 3.0), 0.16666666666666666)
	elif d <= 2.55e-160:
		tmp = t_2
	elif d <= 3.3e-84:
		tmp = t_3
	elif d <= 3.1e-39:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (M * (t_0 / (d / M)))
	elif d <= 6.8e+34:
		tmp = t_3
	elif d <= 2.8e+80:
		tmp = t_2
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * D) * -0.125)
	t_1 = Float64(Float64(1.0 / h) / l)
	t_2 = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(t_0 / Float64(Float64(d / M) / M)))
	t_3 = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)))
	tmp = 0.0
	if (d <= -5.2e-72)
		tmp = Float64(d * Float64(-sqrt(t_1)));
	elseif (d <= -4e-310)
		tmp = Float64(d * ((t_1 ^ 3.0) ^ 0.16666666666666666));
	elseif (d <= 2.55e-160)
		tmp = t_2;
	elseif (d <= 3.3e-84)
		tmp = t_3;
	elseif (d <= 3.1e-39)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(M * Float64(t_0 / Float64(d / M))));
	elseif (d <= 6.8e+34)
		tmp = t_3;
	elseif (d <= 2.8e+80)
		tmp = t_2;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * D) * -0.125;
	t_1 = (1.0 / h) / l;
	t_2 = (sqrt(h) / (l ^ 1.5)) * (t_0 / ((d / M) / M));
	t_3 = d * (sqrt((1.0 / l)) / sqrt(h));
	tmp = 0.0;
	if (d <= -5.2e-72)
		tmp = d * -sqrt(t_1);
	elseif (d <= -4e-310)
		tmp = d * ((t_1 ^ 3.0) ^ 0.16666666666666666);
	elseif (d <= 2.55e-160)
		tmp = t_2;
	elseif (d <= 3.3e-84)
		tmp = t_3;
	elseif (d <= 3.1e-39)
		tmp = sqrt((h / (l ^ 3.0))) * (M * (t_0 / (d / M)));
	elseif (d <= 6.8e+34)
		tmp = t_3;
	elseif (d <= 2.8e+80)
		tmp = t_2;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.2e-72], N[(d * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.55e-160], t$95$2, If[LessEqual[d, 3.3e-84], t$95$3, If[LessEqual[d, 3.1e-39], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(M * N[(t$95$0 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e+34], t$95$3, If[LessEqual[d, 2.8e+80], t$95$2, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -5.19999999999999992e-72

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 71.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 70.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 70.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 70.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 60.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 60.9%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -5.19999999999999992e-72 < d < -3.999999999999988e-310

   1. Initial program 46.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 46.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 46.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 14.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 14.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. pow1/3 35.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot d \]

      4. add-cube-cbrt 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}} \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right) \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}}^{0.3333333333333333}} \cdot d \]

      5. pow3 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot d \]

      6. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      7. *-commutative 35.9%

         \[\leadsto \sqrt{{\left({\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      8. associate-/r* 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

   6. Applied egg-rr 35.9%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot d \]
   7. Step-by-step derivation

      1. sqrt-pow1 35.9%

         \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot d \]

      2. associate-/l/ 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot d \]

      3. metadata-eval 35.9%

         \[\leadsto {\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \cdot d \]

   8. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]
   9. Step-by-step derivation

      1. associate-/r* 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}\right)}^{0.16666666666666666} \cdot d \]

   10. Simplified 35.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]

   if -3.999999999999988e-310 < d < 2.55e-160 or 6.7999999999999999e34 < d < 2.79999999999999984e80

   1. Initial program 55.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 55.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0 51.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 51.2%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 51.2%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 51.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. associate-*r/ 51.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125 \cdot {D}^{2}}{\frac{d}{{M}^{2}}}} \]

      5. unpow2 51.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{d}{{M}^{2}}} \]

      6. unpow2 51.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{\color{blue}{M \cdot M}}} \]

      7. associate-/r* 55.3%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{\frac{d}{M}}{M}}} \]

   8. Simplified 55.3%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}}} \]
   9. Step-by-step derivation

      1. sqrt-div 57.5%

         \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

   10. Applied egg-rr 57.5%

       \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]
   11. Step-by-step derivation

       1. sqr-pow 57.5%

          \[\leadsto \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

       2. rem-sqrt-square 61.9%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

       3. sqr-pow 61.8%

          \[\leadsto \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

       4. fabs-sqr 61.8%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

       5. sqr-pow 61.9%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

       6. metadata-eval 61.9%

          \[\leadsto \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

   12. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}} \]

   if 2.55e-160 < d < 3.29999999999999984e-84 or 3.0999999999999997e-39 < d < 6.7999999999999999e34

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 59.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 59.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 71.0%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   8. Applied egg-rr 71.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   if 3.29999999999999984e-84 < d < 3.0999999999999997e-39

   1. Initial program 88.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 88.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 88.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 88.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 88.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 88.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 88.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 88.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 88.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 88.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 88.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 88.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 88.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 88.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0 56.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 56.4%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 56.4%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 55.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. associate-*r/ 55.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125 \cdot {D}^{2}}{\frac{d}{{M}^{2}}}} \]

      5. unpow2 55.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{d}{{M}^{2}}} \]

      6. unpow2 55.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{\color{blue}{M \cdot M}}} \]

      7. associate-/r* 55.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{\frac{d}{M}}{M}}} \]

   8. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 67.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \]

   10. Applied egg-rr 67.5%

       \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \]

   if 2.79999999999999984e80 < d

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 76.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 76.9%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 57.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 48.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 55.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 55.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 55.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 76.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 77.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 76.9%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 76.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 77.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 77.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 76.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 76.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 77.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 77.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 77.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 81.9%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-72}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-84}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{d}{M}}\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{\frac{d}{M}}{M}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 16: 47.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{h}}{\ell}\\ t_1 := -0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-74}:\\ \;\;\;\;d \cdot \left(-\sqrt{t_0}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({t_0}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-83}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 h) l))
        (t_1
         (* -0.125 (/ (* (sqrt (/ h (pow l 3.0))) (* (* D D) (* M M))) d))))
   (if (<= d -1.7e-74)
     (* d (- (sqrt t_0)))
     (if (<= d -4e-310)
       (* d (pow (pow t_0 3.0) 0.16666666666666666))
       (if (<= d 4.2e-161)
         t_1
         (if (<= d 1.65e-83)
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))
           (if (<= d 5.8e-40) t_1 (* d (* (pow h -0.5) (pow l -0.5))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 / h) / l;
	double t_1 = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((D * D) * (M * M))) / d);
	double tmp;
	if (d <= -1.7e-74) {
		tmp = d * -sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * pow(pow(t_0, 3.0), 0.16666666666666666);
	} else if (d <= 4.2e-161) {
		tmp = t_1;
	} else if (d <= 1.65e-83) {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	} else if (d <= 5.8e-40) {
		tmp = t_1;
	} else {
		tmp = d * (pow(h, -0.5) * pow(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) :: tmp
    t_0 = (1.0d0 / h) / l
    t_1 = (-0.125d0) * ((sqrt((h / (l ** 3.0d0))) * ((d_1 * d_1) * (m * m))) / d)
    if (d <= (-1.7d-74)) then
        tmp = d * -sqrt(t_0)
    else if (d <= (-4d-310)) then
        tmp = d * ((t_0 ** 3.0d0) ** 0.16666666666666666d0)
    else if (d <= 4.2d-161) then
        tmp = t_1
    else if (d <= 1.65d-83) then
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    else if (d <= 5.8d-40) then
        tmp = t_1
    else
        tmp = d * ((h ** (-0.5d0)) * (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 = (1.0 / h) / l;
	double t_1 = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * ((D * D) * (M * M))) / d);
	double tmp;
	if (d <= -1.7e-74) {
		tmp = d * -Math.sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * Math.pow(Math.pow(t_0, 3.0), 0.16666666666666666);
	} else if (d <= 4.2e-161) {
		tmp = t_1;
	} else if (d <= 1.65e-83) {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	} else if (d <= 5.8e-40) {
		tmp = t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 / h) / l
	t_1 = -0.125 * ((math.sqrt((h / math.pow(l, 3.0))) * ((D * D) * (M * M))) / d)
	tmp = 0
	if d <= -1.7e-74:
		tmp = d * -math.sqrt(t_0)
	elif d <= -4e-310:
		tmp = d * math.pow(math.pow(t_0, 3.0), 0.16666666666666666)
	elif d <= 4.2e-161:
		tmp = t_1
	elif d <= 1.65e-83:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	elif d <= 5.8e-40:
		tmp = t_1
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 / h) / l)
	t_1 = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(D * D) * Float64(M * M))) / d))
	tmp = 0.0
	if (d <= -1.7e-74)
		tmp = Float64(d * Float64(-sqrt(t_0)));
	elseif (d <= -4e-310)
		tmp = Float64(d * ((t_0 ^ 3.0) ^ 0.16666666666666666));
	elseif (d <= 4.2e-161)
		tmp = t_1;
	elseif (d <= 1.65e-83)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	elseif (d <= 5.8e-40)
		tmp = t_1;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 / h) / l;
	t_1 = -0.125 * ((sqrt((h / (l ^ 3.0))) * ((D * D) * (M * M))) / d);
	tmp = 0.0;
	if (d <= -1.7e-74)
		tmp = d * -sqrt(t_0);
	elseif (d <= -4e-310)
		tmp = d * ((t_0 ^ 3.0) ^ 0.16666666666666666);
	elseif (d <= 4.2e-161)
		tmp = t_1;
	elseif (d <= 1.65e-83)
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	elseif (d <= 5.8e-40)
		tmp = t_1;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.7e-74], N[(d * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-161], t$95$1, If[LessEqual[d, 1.65e-83], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-40], t$95$1, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.7e-74

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 71.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 70.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 70.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 70.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 60.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 60.9%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -1.7e-74 < d < -3.999999999999988e-310

   1. Initial program 46.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 46.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 46.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 14.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 14.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. pow1/3 35.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot d \]

      4. add-cube-cbrt 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}} \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right) \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}}^{0.3333333333333333}} \cdot d \]

      5. pow3 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot d \]

      6. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      7. *-commutative 35.9%

         \[\leadsto \sqrt{{\left({\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      8. associate-/r* 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

   6. Applied egg-rr 35.9%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot d \]
   7. Step-by-step derivation

      1. sqrt-pow1 35.9%

         \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot d \]

      2. associate-/l/ 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot d \]

      3. metadata-eval 35.9%

         \[\leadsto {\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \cdot d \]

   8. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]
   9. Step-by-step derivation

      1. associate-/r* 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}\right)}^{0.16666666666666666} \cdot d \]

   10. Simplified 35.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]

   if -3.999999999999988e-310 < d < 4.2000000000000001e-161 or 1.65e-83 < d < 5.7999999999999998e-40

   1. Initial program 57.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 57.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 57.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 57.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 57.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 57.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 57.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 57.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around 0 53.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 53.5%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. unpow2 53.5%

         \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

      3. unpow2 53.5%

         \[\leadsto -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   6. Simplified 53.5%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

   if 4.2000000000000001e-161 < d < 1.65e-83

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 48.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 48.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 66.9%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   8. Applied egg-rr 66.9%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   if 5.7999999999999998e-40 < d

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 67.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 47.7%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 47.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 39.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 68.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 67.6%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 67.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 67.7%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 67.7%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 67.6%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 67.6%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 67.7%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 67.7%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 72.1%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 72.1%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-74}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-83}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{h}}{\ell}\\ t_1 := \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{d}{M}}\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;d \cdot \left(-\sqrt{t_0}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({t_0}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 h) l))
        (t_1
         (* (sqrt (/ h (pow l 3.0))) (* M (/ (* (* D D) -0.125) (/ d M))))))
   (if (<= d -2.9e-74)
     (* d (- (sqrt t_0)))
     (if (<= d -4e-310)
       (* d (pow (pow t_0 3.0) 0.16666666666666666))
       (if (<= d 8e-137)
         t_1
         (if (<= d 2.2e-83)
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))
           (if (<= d 9.8e-40) t_1 (* d (* (pow h -0.5) (pow l -0.5))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 / h) / l;
	double t_1 = sqrt((h / pow(l, 3.0))) * (M * (((D * D) * -0.125) / (d / M)));
	double tmp;
	if (d <= -2.9e-74) {
		tmp = d * -sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * pow(pow(t_0, 3.0), 0.16666666666666666);
	} else if (d <= 8e-137) {
		tmp = t_1;
	} else if (d <= 2.2e-83) {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	} else if (d <= 9.8e-40) {
		tmp = t_1;
	} else {
		tmp = d * (pow(h, -0.5) * pow(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) :: tmp
    t_0 = (1.0d0 / h) / l
    t_1 = sqrt((h / (l ** 3.0d0))) * (m * (((d_1 * d_1) * (-0.125d0)) / (d / m)))
    if (d <= (-2.9d-74)) then
        tmp = d * -sqrt(t_0)
    else if (d <= (-4d-310)) then
        tmp = d * ((t_0 ** 3.0d0) ** 0.16666666666666666d0)
    else if (d <= 8d-137) then
        tmp = t_1
    else if (d <= 2.2d-83) then
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    else if (d <= 9.8d-40) then
        tmp = t_1
    else
        tmp = d * ((h ** (-0.5d0)) * (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 = (1.0 / h) / l;
	double t_1 = Math.sqrt((h / Math.pow(l, 3.0))) * (M * (((D * D) * -0.125) / (d / M)));
	double tmp;
	if (d <= -2.9e-74) {
		tmp = d * -Math.sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * Math.pow(Math.pow(t_0, 3.0), 0.16666666666666666);
	} else if (d <= 8e-137) {
		tmp = t_1;
	} else if (d <= 2.2e-83) {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	} else if (d <= 9.8e-40) {
		tmp = t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 / h) / l
	t_1 = math.sqrt((h / math.pow(l, 3.0))) * (M * (((D * D) * -0.125) / (d / M)))
	tmp = 0
	if d <= -2.9e-74:
		tmp = d * -math.sqrt(t_0)
	elif d <= -4e-310:
		tmp = d * math.pow(math.pow(t_0, 3.0), 0.16666666666666666)
	elif d <= 8e-137:
		tmp = t_1
	elif d <= 2.2e-83:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	elif d <= 9.8e-40:
		tmp = t_1
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 / h) / l)
	t_1 = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(M * Float64(Float64(Float64(D * D) * -0.125) / Float64(d / M))))
	tmp = 0.0
	if (d <= -2.9e-74)
		tmp = Float64(d * Float64(-sqrt(t_0)));
	elseif (d <= -4e-310)
		tmp = Float64(d * ((t_0 ^ 3.0) ^ 0.16666666666666666));
	elseif (d <= 8e-137)
		tmp = t_1;
	elseif (d <= 2.2e-83)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	elseif (d <= 9.8e-40)
		tmp = t_1;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 / h) / l;
	t_1 = sqrt((h / (l ^ 3.0))) * (M * (((D * D) * -0.125) / (d / M)));
	tmp = 0.0;
	if (d <= -2.9e-74)
		tmp = d * -sqrt(t_0);
	elseif (d <= -4e-310)
		tmp = d * ((t_0 ^ 3.0) ^ 0.16666666666666666);
	elseif (d <= 8e-137)
		tmp = t_1;
	elseif (d <= 2.2e-83)
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	elseif (d <= 9.8e-40)
		tmp = t_1;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(M * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-74], N[(d * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-137], t$95$1, If[LessEqual[d, 2.2e-83], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.8e-40], t$95$1, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.9e-74

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 71.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 70.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 70.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 70.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 60.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 60.9%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -2.9e-74 < d < -3.999999999999988e-310

   1. Initial program 46.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 46.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 46.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 14.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 14.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. pow1/3 35.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot d \]

      4. add-cube-cbrt 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}} \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right) \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}}^{0.3333333333333333}} \cdot d \]

      5. pow3 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot d \]

      6. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      7. *-commutative 35.9%

         \[\leadsto \sqrt{{\left({\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      8. associate-/r* 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

   6. Applied egg-rr 35.9%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot d \]
   7. Step-by-step derivation

      1. sqrt-pow1 35.9%

         \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot d \]

      2. associate-/l/ 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot d \]

      3. metadata-eval 35.9%

         \[\leadsto {\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \cdot d \]

   8. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]
   9. Step-by-step derivation

      1. associate-/r* 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}\right)}^{0.16666666666666666} \cdot d \]

   10. Simplified 35.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]

   if -3.999999999999988e-310 < d < 7.99999999999999982e-137 or 2.20000000000000008e-83 < d < 9.7999999999999995e-40

   1. Initial program 59.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 59.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 59.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 59.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0 52.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 52.3%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 52.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 52.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. associate-*r/ 52.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125 \cdot {D}^{2}}{\frac{d}{{M}^{2}}}} \]

      5. unpow2 52.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{d}{{M}^{2}}} \]

      6. unpow2 52.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{\color{blue}{M \cdot M}}} \]

      7. associate-/r* 56.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{\frac{d}{M}}{M}}} \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{\frac{d}{M}}{M}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 60.3%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \]

   10. Applied egg-rr 60.3%

       \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \]

   if 7.99999999999999982e-137 < d < 2.20000000000000008e-83

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 52.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 52.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 52.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 52.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 70.1%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   8. Applied egg-rr 70.1%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   if 9.7999999999999995e-40 < d

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 67.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 47.7%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 47.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 39.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 48.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 68.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 67.6%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 67.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 67.7%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 67.7%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 67.6%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 67.6%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 67.7%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 67.7%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 72.1%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 72.1%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{d}{M}}\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 18: 46.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{h}}{\ell}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-84}:\\ \;\;\;\;d \cdot \left(-\sqrt{t_0}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({t_0}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 h) l)))
   (if (<= d -2.1e-84)
     (* d (- (sqrt t_0)))
     (if (<= d -4e-310)
       (* d (pow (pow t_0 3.0) 0.16666666666666666))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 / h) / l;
	double tmp;
	if (d <= -2.1e-84) {
		tmp = d * -sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * pow(pow(t_0, 3.0), 0.16666666666666666);
	} else {
		tmp = d * (pow(h, -0.5) * pow(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) :: tmp
    t_0 = (1.0d0 / h) / l
    if (d <= (-2.1d-84)) then
        tmp = d * -sqrt(t_0)
    else if (d <= (-4d-310)) then
        tmp = d * ((t_0 ** 3.0d0) ** 0.16666666666666666d0)
    else
        tmp = d * ((h ** (-0.5d0)) * (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 = (1.0 / h) / l;
	double tmp;
	if (d <= -2.1e-84) {
		tmp = d * -Math.sqrt(t_0);
	} else if (d <= -4e-310) {
		tmp = d * Math.pow(Math.pow(t_0, 3.0), 0.16666666666666666);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 / h) / l
	tmp = 0
	if d <= -2.1e-84:
		tmp = d * -math.sqrt(t_0)
	elif d <= -4e-310:
		tmp = d * math.pow(math.pow(t_0, 3.0), 0.16666666666666666)
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 / h) / l)
	tmp = 0.0
	if (d <= -2.1e-84)
		tmp = Float64(d * Float64(-sqrt(t_0)));
	elseif (d <= -4e-310)
		tmp = Float64(d * ((t_0 ^ 3.0) ^ 0.16666666666666666));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 / h) / l;
	tmp = 0.0;
	if (d <= -2.1e-84)
		tmp = d * -sqrt(t_0);
	elseif (d <= -4e-310)
		tmp = d * ((t_0 ^ 3.0) ^ 0.16666666666666666);
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -2.1e-84], N[(d * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.09999999999999998e-84

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 71.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 70.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 70.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 70.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 60.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 60.9%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -2.09999999999999998e-84 < d < -3.999999999999988e-310

   1. Initial program 46.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 46.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 46.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 14.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 14.6%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. pow1/3 35.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot d \]

      4. add-cube-cbrt 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}} \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right) \cdot \sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}}^{0.3333333333333333}} \cdot d \]

      5. pow3 35.9%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot d \]

      6. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      7. *-commutative 35.9%

         \[\leadsto \sqrt{{\left({\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot d \]

      8. associate-/r* 35.9%

         \[\leadsto \sqrt{{\left({\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot d \]

   6. Applied egg-rr 35.9%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot d \]
   7. Step-by-step derivation

      1. sqrt-pow1 35.9%

         \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot d \]

      2. associate-/l/ 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot d \]

      3. metadata-eval 35.9%

         \[\leadsto {\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \cdot d \]

   8. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]
   9. Step-by-step derivation

      1. associate-/r* 35.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}\right)}^{0.16666666666666666} \cdot d \]

   10. Simplified 35.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}} \cdot d \]

   if -3.999999999999988e-310 < d

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 52.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 52.1%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 37.0%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 36.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 29.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 37.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 37.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 37.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 51.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 52.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 52.1%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 52.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 52.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 52.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 52.0%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 52.0%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 52.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 52.1%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 52.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 56.6%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 56.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-84}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 19: 45.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 4.8e-204)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.8e-204) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(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 (d <= 4.8d-204) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(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 (d <= 4.8e-204) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 4.8e-204:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 4.8e-204)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 4.8e-204)
		tmp = d * -sqrt(((1.0 / h) / l));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 4.8e-204], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.8e-204

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 60.2%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 60.2%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 60.2%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 60.2%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 40.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 40.6%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 41.0%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 41.0%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 4.8e-204 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 58.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 58.2%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 58.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 58.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 63.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   8. Applied egg-rr 63.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 20: 45.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-203}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.7e-203)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (* d (* (pow h -0.5) (pow l -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.7e-203) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(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 (d <= 2.7d-203) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (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 (d <= 2.7e-203) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.7e-203:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.7e-203)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.7e-203)
		tmp = d * -sqrt(((1.0 / h) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.7e-203], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.69999999999999999e-203

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 60.2%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 60.2%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 60.2%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 60.2%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 40.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 40.6%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 41.0%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 41.0%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 2.69999999999999999e-203 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 58.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 58.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

      2. add-cbrt-cube 41.7%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. expm1-log1p-u 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

      4. expm1-udef 32.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

      5. add-cbrt-cube 40.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

      6. *-commutative 40.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

      7. associate-/r* 40.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 57.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

      2. expm1-log1p 58.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      3. associate-/l/ 58.2%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. unpow-1 58.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      5. sqr-pow 58.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      6. rem-sqrt-square 58.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      7. sqr-pow 58.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 58.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 58.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      10. metadata-eval 58.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 63.6%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 63.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-203}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 21: 40.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -2.15 \cdot 10^{-114}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -2.15e-114) (* d (- t_0)) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -2.15e-114) {
		tmp = d * -t_0;
	} else {
		tmp = d * 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 = sqrt(((1.0d0 / h) / l))
    if (d <= (-2.15d-114)) then
        tmp = d * -t_0
    else
        tmp = d * 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 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -2.15e-114) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if d <= -2.15e-114:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (d <= -2.15e-114)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (d <= -2.15e-114)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.15e-114], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.15e-114

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 71.7%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 71.7%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 71.7%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 71.7%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around -inf 60.0%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.0%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 60.6%

         \[\leadsto -d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -2.15e-114 < d

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 44.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-commutative 44.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 44.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 44.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 36.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{\ell}}{h} \cdot \frac{\frac{1}{\ell}}{h}\right) \cdot \frac{\frac{1}{\ell}}{h}}}} \]

      2. associate-/l/ 36.6%

         \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \frac{\frac{1}{\ell}}{h}\right) \cdot \frac{\frac{1}{\ell}}{h}}} \]

      3. associate-/l/ 36.6%

         \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \color{blue}{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\frac{1}{\ell}}{h}}} \]

      4. associate-/l/ 36.6%

         \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}}} \]

   8. Applied egg-rr 36.6%

      \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 44.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. associate-/r* 44.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Applied egg-rr 44.4%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{-114}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 22: 25.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / 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 * sqrt(((1.0d0 / h) / l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 65.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 65.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 33.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-commutative 33.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. associate-/r* 33.1%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

6. Simplified 33.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 28.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{\ell}}{h} \cdot \frac{\frac{1}{\ell}}{h}\right) \cdot \frac{\frac{1}{\ell}}{h}}}} \]

   2. associate-/l/ 28.7%

      \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \frac{\frac{1}{\ell}}{h}\right) \cdot \frac{\frac{1}{\ell}}{h}}} \]

   3. associate-/l/ 28.7%

      \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \color{blue}{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\frac{1}{\ell}}{h}}} \]

   4. associate-/l/ 28.7%

      \[\leadsto d \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}}} \]

8. Applied egg-rr 28.7%

   \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 33.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   2. associate-/r* 33.1%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

10. Applied egg-rr 33.1%

    \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

11. Final simplification 33.1%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 23: 25.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

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 * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 65.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 65.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 65.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 33.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-commutative 33.0%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   2. add-cbrt-cube 28.7%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

   3. expm1-log1p-u 28.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)\right)} \cdot d \]

   4. expm1-udef 24.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right)} - 1\right)} \cdot d \]

   5. add-cbrt-cube 24.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

   6. *-commutative 24.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)} - 1\right) \cdot d \]

   7. associate-/r* 24.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)} - 1\right) \cdot d \]

6. Applied egg-rr 24.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \cdot d \]
7. Step-by-step derivation

   1. expm1-def 32.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot d \]

   2. expm1-log1p 33.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   3. associate-/l/ 33.0%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. unpow-1 33.0%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   5. sqr-pow 33.0%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   6. rem-sqrt-square 32.6%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   7. sqr-pow 32.5%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   8. fabs-sqr 32.5%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   9. sqr-pow 32.6%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   10. metadata-eval 32.6%

       \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

8. Simplified 32.6%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 32.6%

   \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023227 
(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)))))