Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 75.8%

Time: 24.5s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% 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: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))))
   (if (<= h -1e-309)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (* t_0 (/ (sqrt d) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	double tmp;
	if (h <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else {
		tmp = t_0 * (sqrt(d) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))))
    if (h <= (-1d-309)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else
        tmp = t_0 * (sqrt(d) / sqrt(h))
    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)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	double tmp;
	if (h <= -1e-309) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else {
		tmp = t_0 * (Math.sqrt(d) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))
	tmp = 0
	if h <= -1e-309:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	else:
		tmp = t_0 * (math.sqrt(d) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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)))))
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	else
		tmp = Float64(t_0 * Float64(sqrt(d) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (h <= -1e-309)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	else
		tmp = t_0 * (sqrt(d) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, If[LessEqual[h, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   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. associate-*l* 65.3%

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

      2. metadata-eval 65.3%

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

      3. unpow1/2 65.3%

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

      4. metadata-eval 65.3%

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

      5. unpow1/2 65.3%

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

      6. associate-*l* 65.3%

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

      7. metadata-eval 65.3%

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

      8. times-frac 65.4%

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

   3. Simplified 65.4%

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

      1. frac-2neg 65.4%

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

      2. sqrt-div 76.3%

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

   5. Applied egg-rr 76.3%

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

   if -1.000000000000002e-309 < h

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. metadata-eval 67.2%

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

      3. unpow1/2 67.2%

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

      4. metadata-eval 67.2%

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

      5. unpow1/2 67.2%

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

      6. associate-*l* 67.2%

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

      7. metadata-eval 67.2%

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

      8. times-frac 67.2%

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

   3. Simplified 67.2%

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

      1. sqrt-div 74.2%

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

   5. Applied egg-rr 74.2%

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

4. Final simplification 75.3%

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

Alternative 2: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+165}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - h \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= l -4.4e+25)
     (*
      t_0
      (*
       (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))
       (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= l 6e+165)
       (*
        (/ t_0 (sqrt (/ l d)))
        (- 1.0 (* h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ 0.5 l)))))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (l <= -4.4e+25) {
		tmp = t_0 * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	} else if (l <= 6e+165) {
		tmp = (t_0 / sqrt((l / d))) * (1.0 - (h * (pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / h))
    if (l <= (-4.4d+25)) then
        tmp = t_0 * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * (sqrt(-d) / sqrt(-l)))
    else if (l <= 6d+165) then
        tmp = (t_0 / sqrt((l / d))) * (1.0d0 - (h * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (0.5d0 / 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 t_0 = Math.sqrt((d / h));
	double tmp;
	if (l <= -4.4e+25) {
		tmp = t_0 * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (l <= 6e+165) {
		tmp = (t_0 / Math.sqrt((l / d))) * (1.0 - (h * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / l))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if l <= -4.4e+25:
		tmp = t_0 * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (math.sqrt(-d) / math.sqrt(-l)))
	elif l <= 6e+165:
		tmp = (t_0 / math.sqrt((l / d))) * (1.0 - (h * (math.pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / l))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -4.4e+25)
		tmp = Float64(t_0 * 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 (l <= 6e+165)
		tmp = Float64(Float64(t_0 / sqrt(Float64(l / d))) * Float64(1.0 - Float64(h * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(0.5 / 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)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (l <= -4.4e+25)
		tmp = t_0 * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	elseif (l <= 6e+165)
		tmp = (t_0 / sqrt((l / d))) * (1.0 - (h * (((D * ((M / d) / 2.0)) ^ 2.0) * (0.5 / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	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[l, -4.4e+25], N[(t$95$0 * 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[l, 6e+165], N[(N[(t$95$0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if l < -4.4000000000000001e25

   1. Initial program 59.5%

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

      1. associate-*l* 59.5%

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

      2. metadata-eval 59.5%

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

      3. unpow1/2 59.5%

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

      4. metadata-eval 59.5%

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

      5. unpow1/2 59.5%

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

      6. associate-*l* 59.5%

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

      7. metadata-eval 59.5%

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

      8. times-frac 59.6%

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

   3. Simplified 59.6%

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

      1. frac-2neg 59.6%

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

      2. sqrt-div 65.2%

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

   5. Applied egg-rr 65.2%

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

   if -4.4000000000000001e25 < l < 5.99999999999999981e165

   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 72.5%

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

      8. metadata-eval 72.5%

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

   3. Simplified 72.5%

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

      1. frac-times 73.0%

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

      2. associate-/l* 72.5%

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

      3. *-commutative 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. associate-/r/ 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

      1. clear-num 71.3%

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

      2. sqrt-div 71.7%

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

      3. metadata-eval 71.7%

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

   9. Applied egg-rr 71.7%

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

       1. pow1 71.7%

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

       2. un-div-inv 71.7%

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

       3. *-commutative 71.7%

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

       4. associate-*r/ 71.7%

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

   11. Applied egg-rr 71.7%

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

       1. unpow1 71.7%

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

       2. associate-/l* 71.8%

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

       3. *-commutative 71.8%

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

       4. *-commutative 71.8%

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

   13. Simplified 71.8%

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

       1. expm1-log1p-u 71.4%

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

       2. expm1-udef 71.4%

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

       3. associate-/r/ 71.4%

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

   15. Applied egg-rr 71.4%

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

       1. expm1-def 71.4%

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

       2. expm1-log1p 71.8%

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

       3. associate-*r* 76.5%

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

       4. *-commutative 76.5%

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

       5. *-commutative 76.5%

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

       6. associate-/r* 76.5%

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

   17. Simplified 76.5%

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

   if 5.99999999999999981e165 < l

   1. Initial program 42.6%

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

   2. Taylor expanded in d around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-/r* 48.2%

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

   4. Simplified 48.2%

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

      1. sqrt-div 64.6%

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

   6. Applied egg-rr 64.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - h \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 3: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+165}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.4e+165)
   (*
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
    (sqrt (/ d h)))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.4e+165) {
		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((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 (l <= 2.4d+165) 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((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 (l <= 2.4e+165) {
		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((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.4e+165:
		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((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.4e+165)
		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(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 (l <= 2.4e+165)
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))))) * sqrt((d / h));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.4e+165], 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[(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 l < 2.4e165

   1. Initial program 69.3%

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

      1. associate-*l* 69.3%

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

      2. metadata-eval 69.3%

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

      3. unpow1/2 69.3%

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

      4. metadata-eval 69.3%

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

      5. unpow1/2 69.3%

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

      6. associate-*l* 69.3%

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

      7. metadata-eval 69.3%

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

      8. times-frac 68.9%

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

   3. Simplified 68.9%

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

   if 2.4e165 < l

   1. Initial program 42.6%

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

   2. Taylor expanded in d around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-/r* 48.2%

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

   4. Simplified 48.2%

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

      1. sqrt-div 64.6%

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

   6. Applied egg-rr 64.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+165}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 4: 67.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 3e+165)
   (*
    (- 1.0 (* (pow (* D (/ M (* d 2.0))) 2.0) (* 0.5 (/ h l))))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 3e+165) {
		tmp = (1.0 - (pow((D * (M / (d * 2.0))), 2.0) * (0.5 * (h / l)))) * (sqrt((d / l)) * sqrt((d / h)));
	} 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 (l <= 3d+165) then
        tmp = (1.0d0 - (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) * (0.5d0 * (h / l)))) * (sqrt((d / l)) * sqrt((d / h)))
    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 (l <= 3e+165) {
		tmp = (1.0 - (Math.pow((D * (M / (d * 2.0))), 2.0) * (0.5 * (h / l)))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 3e+165:
		tmp = (1.0 - (math.pow((D * (M / (d * 2.0))), 2.0) * (0.5 * (h / l)))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	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 (l <= 3e+165)
		tmp = Float64(Float64(1.0 - Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) * Float64(0.5 * Float64(h / l)))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	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 (l <= 3e+165)
		tmp = (1.0 - (((D * (M / (d * 2.0))) ^ 2.0) * (0.5 * (h / l)))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 3e+165], N[(N[(1.0 - N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $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 l < 2.9999999999999999e165

   1. Initial program 69.3%

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

      1. metadata-eval 69.3%

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

      2. unpow1/2 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow1/2 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 68.9%

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

      8. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

      1. frac-times 69.3%

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

      2. associate-/l* 68.9%

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

      3. *-commutative 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate-/r/ 68.0%

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

      2. *-commutative 68.0%

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

   7. Simplified 68.0%

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

   if 2.9999999999999999e165 < l

   1. Initial program 42.6%

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

   2. Taylor expanded in d around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-/r* 48.2%

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

   4. Simplified 48.2%

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

      1. sqrt-div 64.6%

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

   6. Applied egg-rr 64.6%

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

4. Final simplification 67.6%

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

Alternative 5: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+159}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.1e+159)
   (*
    (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* 0.5 M)) 2.0))) l))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.1e+159) {
		tmp = (1.0 - ((h * (0.5 * pow(((D / d) * (0.5 * M)), 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	} 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 (l <= 1.1d+159) then
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0))) / l)) * (sqrt((d / l)) * sqrt((d / h)))
    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 (l <= 1.1e+159) {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D / d) * (0.5 * M)), 2.0))) / l)) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.1e+159:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D / d) * (0.5 * M)), 2.0))) / l)) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	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 (l <= 1.1e+159)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	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 (l <= 1.1e+159)
		tmp = (1.0 - ((h * (0.5 * (((D / d) * (0.5 * M)) ^ 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.1e+159], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $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 l < 1.1e159

   1. Initial program 69.3%

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

      1. metadata-eval 69.3%

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

      2. unpow1/2 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow1/2 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 68.9%

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

      8. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

      1. associate-*r* 68.9%

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

      2. frac-times 69.3%

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

      3. *-commutative 69.3%

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

      4. metadata-eval 69.3%

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

      5. associate-*r/ 71.2%

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

      6. metadata-eval 71.2%

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

      7. *-commutative 71.2%

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

      8. frac-times 70.8%

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

      9. div-inv 70.8%

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

      10. metadata-eval 70.8%

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

   5. Applied egg-rr 70.8%

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

   if 1.1e159 < l

   1. Initial program 45.0%

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

   2. Taylor expanded in d around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-/r* 50.0%

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

   4. Simplified 50.0%

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

      1. sqrt-div 64.8%

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

   6. Applied egg-rr 64.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+159}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - h \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.1e+165)
   (*
    (/ (sqrt (/ d h)) (sqrt (/ l d)))
    (- 1.0 (* h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ 0.5 l)))))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.1e+165) {
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - (h * (pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / 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 (l <= 2.1d+165) then
        tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0d0 - (h * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (0.5d0 / 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 (l <= 2.1e+165) {
		tmp = (Math.sqrt((d / h)) / Math.sqrt((l / d))) * (1.0 - (h * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / 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 l <= 2.1e+165:
		tmp = (math.sqrt((d / h)) / math.sqrt((l / d))) * (1.0 - (h * (math.pow((D * ((M / d) / 2.0)), 2.0) * (0.5 / 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 (l <= 2.1e+165)
		tmp = Float64(Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d))) * Float64(1.0 - Float64(h * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(0.5 / 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 (l <= 2.1e+165)
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 - (h * (((D * ((M / d) / 2.0)) ^ 2.0) * (0.5 / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.1e+165], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $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 l < 2.1000000000000001e165

   1. Initial program 69.3%

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

      1. metadata-eval 69.3%

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

      2. unpow1/2 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow1/2 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 68.9%

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

      8. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

      1. frac-times 69.3%

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

      2. associate-/l* 68.9%

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

      3. *-commutative 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate-/r/ 68.0%

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

      2. *-commutative 68.0%

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

   7. Simplified 68.0%

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

      1. clear-num 67.6%

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

      2. sqrt-div 67.9%

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

      3. metadata-eval 67.9%

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

   9. Applied egg-rr 67.9%

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

       1. pow1 67.9%

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

       2. un-div-inv 67.9%

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

       3. *-commutative 67.9%

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

       4. associate-*r/ 67.9%

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

   11. Applied egg-rr 67.9%

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

       1. unpow1 67.9%

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

       2. associate-/l* 67.9%

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

       3. *-commutative 67.9%

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

       4. *-commutative 67.9%

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

   13. Simplified 67.9%

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

       1. expm1-log1p-u 67.5%

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

       2. expm1-udef 67.5%

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

       3. associate-/r/ 67.5%

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

   15. Applied egg-rr 67.5%

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

       1. expm1-def 67.5%

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

       2. expm1-log1p 67.9%

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

       3. associate-*r* 71.3%

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

       4. *-commutative 71.3%

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

       5. *-commutative 71.3%

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

       6. associate-/r* 71.3%

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

   17. Simplified 71.3%

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

   if 2.1000000000000001e165 < l

   1. Initial program 42.6%

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

   2. Taylor expanded in d around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-/r* 48.2%

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

   4. Simplified 48.2%

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

      1. sqrt-div 64.6%

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

   6. Applied egg-rr 64.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - h \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7: 48.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;M \leq 7.8 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot t_0\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\left(D \cdot \left(\left(M \cdot D\right) \cdot \frac{M}{\ell}\right)\right) \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot \frac{0.5}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= M 7.8e-169)
     (* (sqrt (/ d l)) t_0)
     (if (<= M 2.7e-16)
       (*
        (/ t_0 (sqrt (/ l d)))
        (- 1.0 (* 0.125 (* (* D (* (* M D) (/ M l))) (/ h (* d d))))))
       (*
        (sqrt (* (/ d l) (/ d h)))
        (- 1.0 (* (* h (/ 0.5 l)) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (M <= 7.8e-169) {
		tmp = sqrt((d / l)) * t_0;
	} else if (M <= 2.7e-16) {
		tmp = (t_0 / sqrt((l / d))) * (1.0 - (0.125 * ((D * ((M * D) * (M / l))) * (h / (d * d)))));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * pow((M * (0.5 * (D / d))), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / h))
    if (m <= 7.8d-169) then
        tmp = sqrt((d / l)) * t_0
    else if (m <= 2.7d-16) then
        tmp = (t_0 / sqrt((l / d))) * (1.0d0 - (0.125d0 * ((d_1 * ((m * d_1) * (m / l))) * (h / (d * d)))))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h * (0.5d0 / l)) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double tmp;
	if (M <= 7.8e-169) {
		tmp = Math.sqrt((d / l)) * t_0;
	} else if (M <= 2.7e-16) {
		tmp = (t_0 / Math.sqrt((l / d))) * (1.0 - (0.125 * ((D * ((M * D) * (M / l))) * (h / (d * d)))));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * Math.pow((M * (0.5 * (D / d))), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if M <= 7.8e-169:
		tmp = math.sqrt((d / l)) * t_0
	elif M <= 2.7e-16:
		tmp = (t_0 / math.sqrt((l / d))) * (1.0 - (0.125 * ((D * ((M * D) * (M / l))) * (h / (d * d)))))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * math.pow((M * (0.5 * (D / d))), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (M <= 7.8e-169)
		tmp = Float64(sqrt(Float64(d / l)) * t_0);
	elseif (M <= 2.7e-16)
		tmp = Float64(Float64(t_0 / sqrt(Float64(l / d))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(Float64(M * D) * Float64(M / l))) * Float64(h / Float64(d * d))))));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 / l)) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (M <= 7.8e-169)
		tmp = sqrt((d / l)) * t_0;
	elseif (M <= 2.7e-16)
		tmp = (t_0 / sqrt((l / d))) * (1.0 - (0.125 * ((D * ((M * D) * (M / l))) * (h / (d * d)))));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * ((M * (0.5 * (D / d))) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 7.8e-169], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 2.7e-16], N[(N[(t$95$0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D * N[(N[(M * D), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 7.79999999999999954e-169

   1. Initial program 63.7%

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

      1. associate-*l* 63.7%

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

      2. metadata-eval 63.7%

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

      3. unpow1/2 63.7%

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

      4. metadata-eval 63.7%

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

      5. unpow1/2 63.7%

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

      6. sub-neg 63.7%

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

      7. +-commutative 63.7%

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

      8. *-commutative 63.7%

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

      9. distribute-rgt-neg-in 63.7%

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

      10. fma-def 63.7%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in h around 0 44.7%

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

   if 7.79999999999999954e-169 < M < 2.69999999999999999e-16

   1. Initial program 71.5%

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

      1. metadata-eval 71.5%

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

      2. unpow1/2 71.5%

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

      3. metadata-eval 71.5%

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

      4. unpow1/2 71.5%

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

      5. *-commutative 71.5%

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

      6. associate-*l* 71.5%

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

      7. times-frac 71.6%

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

      8. metadata-eval 71.6%

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

   3. Simplified 71.6%

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

      1. frac-times 71.5%

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

      2. associate-/l* 71.6%

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

      3. *-commutative 71.6%

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

   5. Applied egg-rr 71.6%

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

      1. associate-/r/ 71.5%

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

      2. *-commutative 71.5%

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

   7. Simplified 71.5%

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

      1. clear-num 71.5%

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

      2. sqrt-div 73.8%

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

      3. metadata-eval 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. pow1 73.8%

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

       2. un-div-inv 73.8%

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

       3. *-commutative 73.8%

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

       4. associate-*r/ 73.8%

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

   11. Applied egg-rr 73.8%

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

       1. unpow1 73.8%

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

       2. associate-/l* 73.8%

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

       3. *-commutative 73.8%

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

       4. *-commutative 73.8%

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

   13. Simplified 73.8%

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

   14. Taylor expanded in D around 0 59.6%

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

       1. associate-*r/ 59.6%

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

       2. *-commutative 59.6%

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

       3. associate-*r/ 59.6%

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

       4. associate-*r* 56.4%

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

       5. *-commutative 56.4%

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

       6. times-frac 59.5%

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

       7. unpow2 59.5%

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

       8. unpow2 59.5%

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

       9. swap-sqr 63.2%

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

       10. *-commutative 63.2%

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

       11. associate-*r* 62.9%

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

       12. *-commutative 62.9%

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

       13. associate-*r/ 66.0%

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

       14. associate-*l* 69.7%

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

       15. unpow2 69.7%

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

   16. Simplified 69.7%

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

   if 2.69999999999999999e-16 < M

   1. Initial program 69.0%

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

      1. expm1-log1p-u 23.6%

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

      2. expm1-udef 18.7%

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

   3. Applied egg-rr 15.6%

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

      1. expm1-def 19.3%

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

      2. expm1-log1p 59.2%

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

      3. *-commutative 59.2%

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

      4. associate-*r/ 58.0%

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

      5. *-commutative 58.0%

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

      6. *-commutative 58.0%

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

   5. Simplified 58.0%

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

      1. pow1 58.0%

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

      2. *-commutative 58.0%

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

      3. associate-*r/ 58.0%

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

      4. associate-/l* 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. unpow1 59.1%

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

      2. unpow2 59.1%

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

      3. associate-*r* 62.4%

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

      4. associate-/r/ 62.4%

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

      5. associate-*l/ 60.4%

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

      6. *-commutative 60.4%

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

      7. associate-*r* 60.4%

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

      8. associate-*r/ 62.4%

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

      9. associate-*l* 62.4%

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

      10. metadata-eval 62.4%

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

      11. times-frac 62.4%

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

      12. *-lft-identity 62.4%

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

      13. *-commutative 62.4%

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

      14. *-commutative 62.4%

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

      15. associate-/r/ 62.3%

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

      16. associate-*l/ 60.4%

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

      17. *-commutative 60.4%

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

      18. associate-*r* 60.4%

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

      19. associate-*r/ 62.3%

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

      20. associate-*l* 62.3%

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

      21. metadata-eval 62.3%

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

      22. times-frac 62.3%

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

   9. Simplified 59.2%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\left(D \cdot \left(\left(M \cdot D\right) \cdot \frac{M}{\ell}\right)\right) \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot \frac{0.5}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \end{array} \]

Alternative 8: 47.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 2.2e-163)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* d (/ d (* h l))))
    (- 1.0 (* h (* (/ 0.5 l) (pow (* D (/ M (* d 2.0))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.2e-163) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt((d * (d / (h * l)))) * (1.0 - (h * ((0.5 / l) * pow((D * (M / (d * 2.0))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.2d-163) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt((d * (d / (h * l)))) * (1.0d0 - (h * ((0.5d0 / l) * ((d_1 * (m / (d * 2.0d0))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.2e-163) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt((d * (d / (h * l)))) * (1.0 - (h * ((0.5 / l) * Math.pow((D * (M / (d * 2.0))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 2.2e-163:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt((d * (d / (h * l)))) * (1.0 - (h * ((0.5 / l) * math.pow((D * (M / (d * 2.0))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 2.2e-163)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(h * l)))) * Float64(1.0 - Float64(h * Float64(Float64(0.5 / l) * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 2.2e-163)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt((d * (d / (h * l)))) * (1.0 - (h * ((0.5 / l) * ((D * (M / (d * 2.0))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 2.2e-163], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.5 / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.20000000000000011e-163

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

      2. metadata-eval 64.2%

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

      3. unpow1/2 64.2%

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

      4. metadata-eval 64.2%

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

      5. unpow1/2 64.2%

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

      6. sub-neg 64.2%

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

      7. +-commutative 64.2%

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

      8. *-commutative 64.2%

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

      9. distribute-rgt-neg-in 64.2%

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

      10. fma-def 64.2%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in h around 0 45.4%

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

   if 2.20000000000000011e-163 < 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. metadata-eval 69.1%

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

      2. unpow1/2 69.1%

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

      3. metadata-eval 69.1%

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

      4. unpow1/2 69.1%

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

      5. *-commutative 69.1%

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

      6. associate-*l* 69.1%

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

      7. times-frac 70.0%

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

      8. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

      1. frac-times 69.1%

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

      2. associate-/l* 70.0%

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

      3. *-commutative 70.0%

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

   5. Applied egg-rr 70.0%

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

      1. associate-/r/ 69.0%

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

      2. *-commutative 69.0%

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

   7. Simplified 69.0%

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

      1. clear-num 69.1%

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

      2. sqrt-div 69.6%

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

      3. metadata-eval 69.6%

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

   9. Applied egg-rr 69.6%

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

       1. pow1 69.6%

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

       2. un-div-inv 69.7%

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

       3. *-commutative 69.7%

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

       4. associate-*r/ 69.7%

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

   11. Applied egg-rr 69.7%

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

       1. unpow1 69.7%

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

       2. associate-/l* 69.7%

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

       3. *-commutative 69.7%

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

       4. *-commutative 69.7%

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

   13. Simplified 69.7%

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

       1. expm1-log1p-u 30.2%

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

       2. expm1-udef 22.8%

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

       3. sqrt-undiv 17.5%

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

       4. associate-/r/ 17.5%

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

   15. Applied egg-rr 17.5%

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

       1. expm1-def 22.7%

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

       2. expm1-log1p 56.1%

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

       3. associate-/r/ 55.2%

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

       4. *-commutative 55.2%

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

       5. associate-/l/ 49.8%

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

       6. *-commutative 49.8%

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

       7. associate-*r* 50.8%

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

       8. *-commutative 50.8%

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

   17. Simplified 50.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 48.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot \frac{0.5}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 1.4e-165)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (* h (/ 0.5 l)) (pow (* M (* 0.5 (/ D d))) 2.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.4e-165) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * pow((M * (0.5 * (D / d))), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.4d-165) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h * (0.5d0 / l)) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.4e-165) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * Math.pow((M * (0.5 * (D / d))), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 1.4e-165:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * math.pow((M * (0.5 * (D / d))), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 1.4e-165)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 / l)) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 1.4e-165)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * (0.5 / l)) * ((M * (0.5 * (D / d))) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 1.4e-165], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.4e-165

   1. Initial program 63.9%

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

      1. associate-*l* 63.9%

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

      2. metadata-eval 63.9%

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

      3. unpow1/2 63.9%

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

      4. metadata-eval 63.9%

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

      5. unpow1/2 63.9%

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

      6. sub-neg 63.9%

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

      7. +-commutative 63.9%

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

      8. *-commutative 63.9%

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

      9. distribute-rgt-neg-in 63.9%

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

      10. fma-def 63.9%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in h around 0 45.1%

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

   if 1.4e-165 < M

   1. Initial program 69.4%

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

      1. expm1-log1p-u 31.2%

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

      2. expm1-udef 23.0%

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

   3. Applied egg-rr 18.4%

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

      1. expm1-def 23.6%

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

      2. expm1-log1p 56.7%

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

      3. *-commutative 56.7%

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

      4. associate-*r/ 55.8%

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

      5. *-commutative 55.8%

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

      6. *-commutative 55.8%

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

   5. Simplified 55.8%

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

      1. pow1 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*r/ 55.8%

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

      4. associate-/l* 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. unpow1 56.6%

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

      2. unpow2 56.6%

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

      3. associate-*r* 59.9%

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

      4. associate-/r/ 59.9%

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

      5. associate-*l/ 58.5%

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

      6. *-commutative 58.5%

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

      7. associate-*r* 58.5%

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

      8. associate-*r/ 59.9%

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

      9. associate-*l* 59.9%

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

      10. metadata-eval 59.9%

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

      11. times-frac 59.9%

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

      12. *-lft-identity 59.9%

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

      13. *-commutative 59.9%

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

      14. *-commutative 59.9%

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

      15. associate-/r/ 59.9%

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

      16. associate-*l/ 58.5%

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

      17. *-commutative 58.5%

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

      18. associate-*r* 58.5%

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

      19. associate-*r/ 59.9%

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

      20. associate-*l* 59.9%

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

      21. metadata-eval 59.9%

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

      22. times-frac 59.9%

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

   9. Simplified 56.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot \frac{0.5}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \end{array} \]

Alternative 10: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 2.6e-164)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (/ d (/ h (/ d l))))
    (- 1.0 (* (* 0.5 (/ h l)) (pow (/ (* D (* 0.5 M)) d) 2.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.6e-164) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt((d / (h / (d / l)))) * (1.0 - ((0.5 * (h / l)) * pow(((D * (0.5 * M)) / d), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.6d-164) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt((d / (h / (d / l)))) * (1.0d0 - ((0.5d0 * (h / l)) * (((d_1 * (0.5d0 * m)) / d) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.6e-164) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt((d / (h / (d / l)))) * (1.0 - ((0.5 * (h / l)) * Math.pow(((D * (0.5 * M)) / d), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 2.6e-164:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt((d / (h / (d / l)))) * (1.0 - ((0.5 * (h / l)) * math.pow(((D * (0.5 * M)) / d), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 2.6e-164)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(d / Float64(h / Float64(d / l)))) * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 2.6e-164)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt((d / (h / (d / l)))) * (1.0 - ((0.5 * (h / l)) * (((D * (0.5 * M)) / d) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 2.6e-164], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.6000000000000002e-164

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

      2. metadata-eval 64.2%

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

      3. unpow1/2 64.2%

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

      4. metadata-eval 64.2%

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

      5. unpow1/2 64.2%

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

      6. sub-neg 64.2%

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

      7. +-commutative 64.2%

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

      8. *-commutative 64.2%

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

      9. distribute-rgt-neg-in 64.2%

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

      10. fma-def 64.2%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in h around 0 45.4%

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

   if 2.6000000000000002e-164 < 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. expm1-log1p-u 30.6%

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

      2. expm1-udef 22.4%

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

   3. Applied egg-rr 17.7%

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

      1. expm1-def 22.9%

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

      2. expm1-log1p 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-*r/ 55.4%

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

      5. *-commutative 55.4%

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

      6. *-commutative 55.4%

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

   5. Simplified 55.4%

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

      1. *-un-lft-identity 55.4%

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

      2. *-commutative 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. *-lft-identity 55.4%

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

      2. associate-*l/ 52.6%

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

      3. associate-/l* 54.5%

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

   9. Simplified 54.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 11: 40.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{if}\;M \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))
   (if (<= M 7.5e-77)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= M 1.25e-46)
       t_0
       (if (<= M 7.2e-7)
         (* d (sqrt (/ (/ 1.0 h) l)))
         (if (<= M 4.5e+15)
           (* -0.125 (* (sqrt (/ h (pow l 3.0))) (/ (* D D) (/ d (* M M)))))
           t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	double tmp;
	if (M <= 7.5e-77) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (M <= 1.25e-46) {
		tmp = t_0;
	} else if (M <= 7.2e-7) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else if (M <= 4.5e+15) {
		tmp = -0.125 * (sqrt((h / pow(l, 3.0))) * ((D * D) / (d / (M * M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * ((m * m) / (l / h))))
    if (m <= 7.5d-77) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (m <= 1.25d-46) then
        tmp = t_0
    else if (m <= 7.2d-7) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else if (m <= 4.5d+15) then
        tmp = (-0.125d0) * (sqrt((h / (l ** 3.0d0))) * ((d_1 * d_1) / (d / (m * m))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	double tmp;
	if (M <= 7.5e-77) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (M <= 1.25e-46) {
		tmp = t_0;
	} else if (M <= 7.2e-7) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else if (M <= 4.5e+15) {
		tmp = -0.125 * (Math.sqrt((h / Math.pow(l, 3.0))) * ((D * D) / (d / (M * M))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))))
	tmp = 0
	if M <= 7.5e-77:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif M <= 1.25e-46:
		tmp = t_0
	elif M <= 7.2e-7:
		tmp = d * math.sqrt(((1.0 / h) / l))
	elif M <= 4.5e+15:
		tmp = -0.125 * (math.sqrt((h / math.pow(l, 3.0))) * ((D * D) / (d / (M * M))))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))
	tmp = 0.0
	if (M <= 7.5e-77)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (M <= 1.25e-46)
		tmp = t_0;
	elseif (M <= 7.2e-7)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (M <= 4.5e+15)
		tmp = Float64(-0.125 * Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(D * D) / Float64(d / Float64(M * M)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	tmp = 0.0;
	if (M <= 7.5e-77)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (M <= 1.25e-46)
		tmp = t_0;
	elseif (M <= 7.2e-7)
		tmp = d * sqrt(((1.0 / h) / l));
	elseif (M <= 4.5e+15)
		tmp = -0.125 * (sqrt((h / (l ^ 3.0))) * ((D * D) / (d / (M * M))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 7.5e-77], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.25e-46], t$95$0, If[LessEqual[M, 7.2e-7], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 4.5e+15], N[(-0.125 * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if M < 7.5000000000000006e-77

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. metadata-eval 64.6%

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

      3. unpow1/2 64.6%

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

      4. metadata-eval 64.6%

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

      5. unpow1/2 64.6%

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

      6. sub-neg 64.6%

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

      7. +-commutative 64.6%

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

      8. *-commutative 64.6%

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

      9. distribute-rgt-neg-in 64.6%

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

      10. fma-def 64.6%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in h around 0 46.9%

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

   if 7.5000000000000006e-77 < M < 1.24999999999999998e-46 or 4.5e15 < M

   1. Initial program 71.2%

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

      1. expm1-log1p-u 22.0%

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

      2. expm1-udef 18.2%

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

   3. Applied egg-rr 13.8%

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

      1. expm1-def 16.2%

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

      2. expm1-log1p 59.4%

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

      3. *-commutative 59.4%

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

      4. sub-neg 59.4%

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

      5. +-commutative 59.4%

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

      6. distribute-rgt-neg-in 59.4%

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

      7. fma-def 59.4%

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

      8. associate-*r/ 58.2%

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

      9. associate-*l* 58.2%

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

      10. associate-/l* 59.4%

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

      11. *-commutative 59.4%

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

      12. distribute-rgt-neg-in 59.4%

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

      13. metadata-eval 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in M around inf 28.5%

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

      1. times-frac 27.2%

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

      2. unpow2 27.2%

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

      3. unpow2 27.2%

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

      4. times-frac 35.5%

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

      5. associate-/l* 36.7%

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

      6. unpow2 36.7%

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

   8. Simplified 36.7%

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

   if 1.24999999999999998e-46 < M < 7.19999999999999989e-7

   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. Taylor expanded in d around inf 18.2%

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

      1. *-commutative 18.2%

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

      2. associate-/r* 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in d around 0 18.2%

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

      1. *-commutative 18.2%

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

      2. associate-/l/ 18.2%

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

   7. Simplified 18.2%

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

   if 7.19999999999999989e-7 < M < 4.5e15

   1. Initial program 69.6%

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

   2. Taylor expanded in d around 0 31.8%

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

      1. associate-/l* 31.8%

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

      2. unpow2 31.8%

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

      3. unpow2 31.8%

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

   4. Simplified 31.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;M \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 12: 40.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{if}\;M \leq 6.6 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))
   (if (<= M 6.6e-77)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= M 1.1e-47)
       t_0
       (if (<= M 1.5e-8)
         (* d (sqrt (/ (/ 1.0 h) l)))
         (if (<= M 2.3e+15)
           (* -0.125 (/ (* D (* M (* M D))) (/ d (sqrt (/ h (pow l 3.0))))))
           t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	double tmp;
	if (M <= 6.6e-77) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (M <= 1.1e-47) {
		tmp = t_0;
	} else if (M <= 1.5e-8) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else if (M <= 2.3e+15) {
		tmp = -0.125 * ((D * (M * (M * D))) / (d / sqrt((h / pow(l, 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * ((m * m) / (l / h))))
    if (m <= 6.6d-77) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (m <= 1.1d-47) then
        tmp = t_0
    else if (m <= 1.5d-8) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else if (m <= 2.3d+15) then
        tmp = (-0.125d0) * ((d_1 * (m * (m * d_1))) / (d / sqrt((h / (l ** 3.0d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	double tmp;
	if (M <= 6.6e-77) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (M <= 1.1e-47) {
		tmp = t_0;
	} else if (M <= 1.5e-8) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else if (M <= 2.3e+15) {
		tmp = -0.125 * ((D * (M * (M * D))) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))))
	tmp = 0
	if M <= 6.6e-77:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif M <= 1.1e-47:
		tmp = t_0
	elif M <= 1.5e-8:
		tmp = d * math.sqrt(((1.0 / h) / l))
	elif M <= 2.3e+15:
		tmp = -0.125 * ((D * (M * (M * D))) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))
	tmp = 0.0
	if (M <= 6.6e-77)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (M <= 1.1e-47)
		tmp = t_0;
	elseif (M <= 1.5e-8)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (M <= 2.3e+15)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(M * Float64(M * D))) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	tmp = 0.0;
	if (M <= 6.6e-77)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (M <= 1.1e-47)
		tmp = t_0;
	elseif (M <= 1.5e-8)
		tmp = d * sqrt(((1.0 / h) / l));
	elseif (M <= 2.3e+15)
		tmp = -0.125 * ((D * (M * (M * D))) / (d / sqrt((h / (l ^ 3.0)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 6.6e-77], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.1e-47], t$95$0, If[LessEqual[M, 1.5e-8], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.3e+15], N[(-0.125 * N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if M < 6.59999999999999982e-77

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. metadata-eval 64.6%

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

      3. unpow1/2 64.6%

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

      4. metadata-eval 64.6%

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

      5. unpow1/2 64.6%

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

      6. sub-neg 64.6%

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

      7. +-commutative 64.6%

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

      8. *-commutative 64.6%

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

      9. distribute-rgt-neg-in 64.6%

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

      10. fma-def 64.6%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in h around 0 46.9%

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

   if 6.59999999999999982e-77 < M < 1.10000000000000009e-47 or 2.3e15 < M

   1. Initial program 71.2%

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

      1. expm1-log1p-u 22.0%

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

      2. expm1-udef 18.2%

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

   3. Applied egg-rr 13.8%

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

      1. expm1-def 16.2%

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

      2. expm1-log1p 59.4%

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

      3. *-commutative 59.4%

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

      4. sub-neg 59.4%

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

      5. +-commutative 59.4%

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

      6. distribute-rgt-neg-in 59.4%

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

      7. fma-def 59.4%

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

      8. associate-*r/ 58.2%

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

      9. associate-*l* 58.2%

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

      10. associate-/l* 59.4%

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

      11. *-commutative 59.4%

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

      12. distribute-rgt-neg-in 59.4%

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

      13. metadata-eval 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in M around inf 28.5%

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

      1. times-frac 27.2%

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

      2. unpow2 27.2%

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

      3. unpow2 27.2%

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

      4. times-frac 35.5%

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

      5. associate-/l* 36.7%

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

      6. unpow2 36.7%

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

   8. Simplified 36.7%

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

   if 1.10000000000000009e-47 < M < 1.49999999999999987e-8

   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. Taylor expanded in d around inf 18.2%

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

      1. *-commutative 18.2%

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

      2. associate-/r* 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in d around 0 18.2%

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

      1. *-commutative 18.2%

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

      2. associate-/l/ 18.2%

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

   7. Simplified 18.2%

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

   if 1.49999999999999987e-8 < M < 2.3e15

   1. Initial program 69.6%

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

   2. Taylor expanded in d around 0 31.8%

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

      1. associate-*l/ 31.8%

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

      2. *-commutative 31.8%

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

      3. unpow2 31.8%

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

      4. unpow2 31.8%

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

      5. swap-sqr 31.8%

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

      6. associate-/l* 31.9%

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

      7. *-commutative 31.9%

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

      8. associate-*l* 31.9%

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

      9. *-commutative 31.9%

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

   4. Simplified 31.9%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.6 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 13: 47.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-169}:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-206}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-286}:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -1.85e-118)
     (* d (- (sqrt (/ (/ 1.0 l) h))))
     (if (<= l -2.7e-169)
       (* t_0 (* -0.125 (* (/ (* D D) (* d d)) (/ (* h (* M M)) l))))
       (if (<= l -8e-206)
         (* d (- (pow (* h l) -0.5)))
         (if (<= l 8.5e-286)
           (* t_0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -1.85e-118) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -2.7e-169) {
		tmp = t_0 * (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l)));
	} else if (l <= -8e-206) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= 8.5e-286) {
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-1.85d-118)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= (-2.7d-169)) then
        tmp = t_0 * ((-0.125d0) * (((d_1 * d_1) / (d * d)) * ((h * (m * m)) / l)))
    else if (l <= (-8d-206)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (l <= 8.5d-286) then
        tmp = t_0 * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * ((m * m) / (l / h))))
    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 t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -1.85e-118) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2.7e-169) {
		tmp = t_0 * (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l)));
	} else if (l <= -8e-206) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= 8.5e-286) {
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -1.85e-118:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -2.7e-169:
		tmp = t_0 * (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l)))
	elif l <= -8e-206:
		tmp = d * -math.pow((h * l), -0.5)
	elif l <= 8.5e-286:
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -1.85e-118)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -2.7e-169)
		tmp = Float64(t_0 * Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l))));
	elseif (l <= -8e-206)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= 8.5e-286)
		tmp = Float64(t_0 * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))));
	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)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -1.85e-118)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= -2.7e-169)
		tmp = t_0 * (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l)));
	elseif (l <= -8e-206)
		tmp = d * -((h * l) ^ -0.5);
	elseif (l <= 8.5e-286)
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.85e-118], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2.7e-169], N[(t$95$0 * N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8e-206], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 8.5e-286], N[(t$95$0 * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $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 5 regimes

2. if l < -1.85000000000000007e-118

   1. Initial program 63.9%

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

      1. metadata-eval 63.9%

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

      2. unpow1/2 63.9%

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

      3. metadata-eval 63.9%

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

      4. unpow1/2 63.9%

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

      5. *-commutative 63.9%

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

      6. associate-*l* 63.9%

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

      7. times-frac 64.0%

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

      8. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

      1. frac-times 63.9%

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

      2. associate-/l* 64.0%

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

      3. *-commutative 64.0%

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

   5. Applied egg-rr 64.0%

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

      1. associate-/r/ 63.9%

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

      2. *-commutative 63.9%

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

   7. Simplified 63.9%

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

      1. clear-num 62.8%

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

      2. sqrt-div 62.7%

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

      3. metadata-eval 62.7%

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

   9. Applied egg-rr 62.7%

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

   10. Taylor expanded in d around -inf 49.4%

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

       1. mul-1-neg 49.4%

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

       2. distribute-rgt-neg-in 49.4%

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

       3. associate-/r* 50.5%

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

   12. Simplified 50.5%

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

   if -1.85000000000000007e-118 < l < -2.7000000000000002e-169

   1. Initial program 64.3%

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

      1. expm1-log1p-u 7.1%

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

      2. expm1-udef 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. expm1-def 7.1%

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

      2. expm1-log1p 64.3%

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

      3. *-commutative 64.3%

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

      4. sub-neg 64.3%

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

      5. +-commutative 64.3%

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

      6. distribute-rgt-neg-in 64.3%

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

      7. fma-def 64.3%

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

      8. associate-*r/ 64.3%

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

      9. associate-*l* 64.3%

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

      10. associate-/l* 64.3%

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

      11. *-commutative 64.3%

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

      12. distribute-rgt-neg-in 64.3%

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

      13. metadata-eval 64.3%

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

   5. Simplified 64.3%

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

   6. Taylor expanded in M around inf 64.4%

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

      1. times-frac 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. unpow2 50.4%

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

   8. Simplified 50.4%

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

   if -2.7000000000000002e-169 < l < -8.00000000000000023e-206

   1. Initial program 87.5%

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

      1. metadata-eval 87.5%

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

      2. unpow1/2 87.5%

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

      3. metadata-eval 87.5%

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

      4. unpow1/2 87.5%

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

      5. *-commutative 87.5%

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

      6. associate-*l* 87.5%

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

      7. times-frac 87.5%

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

      8. metadata-eval 87.5%

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

   3. Simplified 87.5%

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

      1. frac-times 87.5%

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

      2. associate-/l* 87.5%

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

      3. *-commutative 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-/r/ 87.5%

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

      2. *-commutative 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num 87.5%

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

      2. sqrt-div 87.5%

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

      3. metadata-eval 87.5%

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

   9. Applied egg-rr 87.5%

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

       1. pow1 87.5%

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

       2. un-div-inv 87.5%

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

       3. *-commutative 87.5%

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

       4. associate-*r/ 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. unpow1 87.5%

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

       2. associate-/l* 87.5%

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

       3. *-commutative 87.5%

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

       4. *-commutative 87.5%

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

   13. Simplified 87.5%

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

   14. Taylor expanded in d around -inf 75.7%

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

       1. mul-1-neg 75.7%

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

       2. distribute-rgt-neg-in 75.7%

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

       3. unpow-1 75.7%

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

       4. sqr-pow 75.7%

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

       5. rem-sqrt-square 75.7%

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

       6. sqr-pow 75.5%

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

       7. fabs-sqr 75.5%

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

       8. sqr-pow 75.7%

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

       9. *-commutative 75.7%

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

       10. metadata-eval 75.7%

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

   16. Simplified 75.7%

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

   if -8.00000000000000023e-206 < l < 8.4999999999999998e-286

   1. Initial program 66.9%

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

      1. expm1-log1p-u 4.0%

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

      2. expm1-udef 4.0%

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

   3. Applied egg-rr 0.6%

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

      1. expm1-def 0.6%

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

      2. expm1-log1p 63.0%

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

      3. *-commutative 63.0%

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

      4. sub-neg 63.0%

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

      5. +-commutative 63.0%

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

      6. distribute-rgt-neg-in 63.0%

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

      7. fma-def 63.0%

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

      8. associate-*r/ 63.0%

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

      9. associate-*l* 63.0%

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

      10. associate-/l* 63.0%

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

      11. *-commutative 63.0%

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

      12. distribute-rgt-neg-in 63.0%

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

      13. metadata-eval 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in M around inf 37.5%

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

      1. times-frac 41.7%

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

      2. unpow2 41.7%

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

      3. unpow2 41.7%

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

      4. times-frac 58.3%

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

      5. associate-/l* 58.3%

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

      6. unpow2 58.3%

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

   8. Simplified 58.3%

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

   if 8.4999999999999998e-286 < l

   1. Initial program 66.7%

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

   2. Taylor expanded in d around inf 41.9%

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

      1. *-commutative 41.9%

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

      2. associate-/r* 41.9%

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

   4. Simplified 41.9%

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

      1. sqrt-div 50.4%

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

   6. Applied egg-rr 50.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-206}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14: 40.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 6.6 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 6.6e-77)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 6.6e-77) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (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 (m <= 6.6d-77) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * ((m * m) / (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 (M <= 6.6e-77) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 6.6e-77:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 6.6e-77)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 6.6e-77)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 6.6e-77], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.59999999999999982e-77

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. metadata-eval 64.6%

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

      3. unpow1/2 64.6%

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

      4. metadata-eval 64.6%

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

      5. unpow1/2 64.6%

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

      6. sub-neg 64.6%

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

      7. +-commutative 64.6%

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

      8. *-commutative 64.6%

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

      9. distribute-rgt-neg-in 64.6%

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

      10. fma-def 64.6%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in h around 0 46.9%

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

   if 6.59999999999999982e-77 < M

   1. Initial program 69.2%

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

      1. expm1-log1p-u 25.6%

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

      2. expm1-udef 19.4%

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

   3. Applied egg-rr 14.9%

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

      1. expm1-def 18.9%

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

      2. expm1-log1p 56.5%

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

      3. *-commutative 56.5%

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

      4. sub-neg 56.5%

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

      5. +-commutative 56.5%

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

      6. distribute-rgt-neg-in 56.5%

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

      7. fma-def 56.5%

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

      8. associate-*r/ 55.4%

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

      9. associate-*l* 55.4%

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

      10. associate-/l* 56.4%

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

      11. *-commutative 56.4%

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

      12. distribute-rgt-neg-in 56.4%

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

      13. metadata-eval 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in M around inf 23.4%

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

      1. times-frac 23.4%

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

      2. unpow2 23.4%

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

      3. unpow2 23.4%

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

      4. times-frac 31.4%

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

      5. associate-/l* 32.5%

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

      6. unpow2 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.6 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 15: 35.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;M \leq 1.1 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 2.45 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= M 1.1e-213)
     t_0
     (if (<= M 5.4e-197)
       (* d (sqrt (/ (/ 1.0 h) l)))
       (if (<= M 2.45e-77)
         t_0
         (* t_0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (M <= 1.1e-213) {
		tmp = t_0;
	} else if (M <= 5.4e-197) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else if (M <= 2.45e-77) {
		tmp = t_0;
	} else {
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (m <= 1.1d-213) then
        tmp = t_0
    else if (m <= 5.4d-197) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else if (m <= 2.45d-77) then
        tmp = t_0
    else
        tmp = t_0 * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * ((m * m) / (l / h))))
    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) * (d / h)));
	double tmp;
	if (M <= 1.1e-213) {
		tmp = t_0;
	} else if (M <= 5.4e-197) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else if (M <= 2.45e-77) {
		tmp = t_0;
	} else {
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if M <= 1.1e-213:
		tmp = t_0
	elif M <= 5.4e-197:
		tmp = d * math.sqrt(((1.0 / h) / l))
	elif M <= 2.45e-77:
		tmp = t_0
	else:
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (M <= 1.1e-213)
		tmp = t_0;
	elseif (M <= 5.4e-197)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (M <= 2.45e-77)
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (M <= 1.1e-213)
		tmp = t_0;
	elseif (M <= 5.4e-197)
		tmp = d * sqrt(((1.0 / h) / l));
	elseif (M <= 2.45e-77)
		tmp = t_0;
	else
		tmp = t_0 * (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 1.1e-213], t$95$0, If[LessEqual[M, 5.4e-197], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.45e-77], t$95$0, N[(t$95$0 * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.10000000000000005e-213 or 5.40000000000000034e-197 < M < 2.4499999999999999e-77

   1. Initial program 65.1%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 30.2%

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

   3. Applied egg-rr 24.3%

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

      1. expm1-def 31.0%

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

      2. expm1-log1p 52.2%

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

      3. *-commutative 52.2%

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

      4. sub-neg 52.2%

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

      5. +-commutative 52.2%

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

      6. distribute-rgt-neg-in 52.2%

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

      7. fma-def 52.2%

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

      8. associate-*r/ 52.9%

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

      9. associate-*l* 52.9%

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

      10. associate-/l* 52.3%

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

      11. *-commutative 52.3%

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

      12. distribute-rgt-neg-in 52.3%

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

      13. metadata-eval 52.3%

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

   5. Simplified 52.3%

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

   6. Taylor expanded in M around 0 35.9%

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

   if 1.10000000000000005e-213 < M < 5.40000000000000034e-197

   1. Initial program 53.0%

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

   2. Taylor expanded in d around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/r* 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/l/ 69.5%

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

   7. Simplified 69.5%

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

   if 2.4499999999999999e-77 < M

   1. Initial program 69.2%

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

      1. expm1-log1p-u 25.6%

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

      2. expm1-udef 19.4%

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

   3. Applied egg-rr 14.9%

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

      1. expm1-def 18.9%

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

      2. expm1-log1p 56.5%

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

      3. *-commutative 56.5%

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

      4. sub-neg 56.5%

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

      5. +-commutative 56.5%

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

      6. distribute-rgt-neg-in 56.5%

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

      7. fma-def 56.5%

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

      8. associate-*r/ 55.4%

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

      9. associate-*l* 55.4%

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

      10. associate-/l* 56.4%

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

      11. *-commutative 56.4%

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

      12. distribute-rgt-neg-in 56.4%

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

      13. metadata-eval 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in M around inf 23.4%

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

      1. times-frac 23.4%

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

      2. unpow2 23.4%

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

      3. unpow2 23.4%

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

      4. times-frac 31.4%

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

      5. associate-/l* 32.5%

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

      6. unpow2 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;M \leq 5.4 \cdot 10^{-197}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;M \leq 2.45 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 16: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.18e-216)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.18e-216) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * sqrt(((1.0 / h) / 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 (l <= (-1.18d-216)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * sqrt(((1.0d0 / h) / 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 (l <= -1.18e-216) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.18e-216:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.18e-216)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.18e-216)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.18e-216], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.18e-216

   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. Step-by-step derivation

      1. frac-times 65.7%

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

      2. associate-/l* 65.7%

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

      3. *-commutative 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. associate-/r/ 65.7%

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

      2. *-commutative 65.7%

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

   7. Simplified 65.7%

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

      1. clear-num 64.8%

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

      2. sqrt-div 64.7%

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

      3. metadata-eval 64.7%

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

   9. Applied egg-rr 64.7%

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

   10. Taylor expanded in d around -inf 47.0%

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

       1. mul-1-neg 47.0%

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

       2. distribute-rgt-neg-in 47.0%

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

       3. associate-/r* 47.8%

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

   12. Simplified 47.8%

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

   if -1.18e-216 < l

   1. Initial program 66.7%

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

   2. Taylor expanded in d around inf 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-/r* 38.7%

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

   4. Simplified 38.7%

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

   5. Taylor expanded in d around 0 38.7%

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

      1. *-commutative 38.7%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 38.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 38.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 17: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= d 5.8e-204) (* d (- t_0)) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (d <= 5.8e-204) {
		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 = (h * l) ** (-0.5d0)
    if (d <= 5.8d-204) 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.pow((h * l), -0.5);
	double tmp;
	if (d <= 5.8e-204) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if d <= 5.8e-204:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (d <= 5.8e-204)
		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 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= 5.8e-204)
		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[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, 5.8e-204], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 5.80000000000000018e-204

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/l* 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{\frac{\color{blue}{d \cdot 2}}{D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 62.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{d \cdot 2} \cdot D\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{\color{blue}{2 \cdot d}} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. clear-num 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 60.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 60.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   9. Applied egg-rr 60.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   10. Step-by-step derivation

       1. pow1 60.9%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

       2. un-div-inv 60.9%

          \[\leadsto {\left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

       3. *-commutative 60.9%

          \[\leadsto {\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{\color{blue}{d \cdot 2}} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

       4. associate-*r/ 60.9%

          \[\leadsto {\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right)\right)}^{1} \]

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 60.9%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

       2. associate-/l* 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

       3. *-commutative 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       4. *-commutative 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   13. Simplified 61.0%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   14. Taylor expanded in d around -inf 38.3%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   15. Step-by-step derivation

       1. mul-1-neg 38.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. distribute-rgt-neg-in 38.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

       3. unpow-1 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

       4. sqr-pow 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

       5. rem-sqrt-square 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

       6. sqr-pow 38.2%

          \[\leadsto d \cdot \left(-\left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right|\right) \]

       7. fabs-sqr 38.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right) \]

       8. sqr-pow 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

       9. *-commutative 38.3%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

       10. metadata-eval 38.3%

           \[\leadsto d \cdot \left(-{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   16. Simplified 38.3%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 5.80000000000000018e-204 < d

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 47.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

      2. expm1-udef 26.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]

   4. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]
   5. Step-by-step derivation

      1. expm1-def 47.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

      2. expm1-log1p 48.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      3. unpow-1 48.0%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      4. sqr-pow 48.0%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 48.0%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. sqr-pow 47.8%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 47.8%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 48.0%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      9. *-commutative 48.0%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{\left(\frac{-1}{2}\right)} \cdot d \]

      10. metadata-eval 48.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 48.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 18: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.16 \cdot 10^{-200}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.16e-200)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.16e-200) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / 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 <= 1.16d-200) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / 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 <= 1.16e-200) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.16e-200:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.16e-200)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.16e-200)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.16e-200], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.1600000000000001e-200

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/l* 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{\frac{\color{blue}{d \cdot 2}}{D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 62.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{d \cdot 2} \cdot D\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 61.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{\color{blue}{2 \cdot d}} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. clear-num 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 60.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 60.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   9. Applied egg-rr 60.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   10. Step-by-step derivation

       1. pow1 60.9%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

       2. un-div-inv 60.9%

          \[\leadsto {\left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - {\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

       3. *-commutative 60.9%

          \[\leadsto {\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{\color{blue}{d \cdot 2}} \cdot D\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

       4. associate-*r/ 60.9%

          \[\leadsto {\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right)\right)}^{1} \]

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 60.9%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

       2. associate-/l* 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

       3. *-commutative 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       4. *-commutative 61.0%

          \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   13. Simplified 61.0%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   14. Taylor expanded in d around -inf 38.3%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   15. Step-by-step derivation

       1. mul-1-neg 38.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. distribute-rgt-neg-in 38.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

       3. unpow-1 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

       4. sqr-pow 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

       5. rem-sqrt-square 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

       6. sqr-pow 38.2%

          \[\leadsto d \cdot \left(-\left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right|\right) \]

       7. fabs-sqr 38.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right) \]

       8. sqr-pow 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

       9. *-commutative 38.3%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

       10. metadata-eval 38.3%

           \[\leadsto d \cdot \left(-{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   16. Simplified 38.3%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 1.1600000000000001e-200 < d

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 48.0%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 48.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   4. Simplified 48.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

   5. Taylor expanded in d around 0 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

      1. *-commutative 48.0%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/l/ 48.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 48.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.16 \cdot 10^{-200}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 19: 27.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -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 * ((h * l) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in d around inf 24.9%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Step-by-step derivation

   1. expm1-log1p-u 24.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

   2. expm1-udef 16.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]

4. Applied egg-rr 16.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]
5. Step-by-step derivation

   1. expm1-def 24.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

   2. expm1-log1p 24.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

   3. unpow-1 24.9%

      \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

   4. sqr-pow 24.9%

      \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   5. rem-sqrt-square 24.2%

      \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   6. sqr-pow 24.1%

      \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   7. fabs-sqr 24.1%

      \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   8. sqr-pow 24.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   9. *-commutative 24.2%

      \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{\left(\frac{-1}{2}\right)} \cdot d \]

   10. metadata-eval 24.2%

       \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

6. Simplified 24.2%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

7. Final simplification 24.2%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023230 
(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)))))