Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 80.4%

Time: 22.7s

Alternatives: 14

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: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5)))))
   (if (<= l -2e-310)
     (* (/ (sqrt (- d)) (sqrt (- h))) (* (sqrt (/ d l)) t_0))
     (* (/ (sqrt d) (sqrt h)) (* t_0 (/ (sqrt d) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5));
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))
    if (l <= (-2d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5));
	double tmp;
	if (l <= -2e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * t_0);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))
	tmp = 0
	if l <= -2e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * t_0)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 70.7%

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

   3. Simplified 69.9%

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

      1. frac-2neg 69.9%

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

      2. sqrt-div 82.5%

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

   5. Applied egg-rr 82.5%

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

   if -1.999999999999994e-310 < l

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. sqrt-div 84.4%

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

      2. div-inv 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

   7. Simplified 84.4%

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

      1. sqrt-div 87.1%

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

      2. div-inv 87.1%

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

   9. Applied egg-rr 87.1%

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

       1. associate-*r/ 87.1%

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

       2. *-rgt-identity 87.1%

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

   11. Simplified 87.1%

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

4. Final simplification 84.9%

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

Alternative 2: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -2.5e+150)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l 8.8e-275)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* h (* (pow (* M (* D (/ 0.5 d))) 2.0) (/ -0.5 l))))))
       (*
        (* t_0 (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))))
        (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -2.5e+150) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= 8.8e-275) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * (pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))));
	} else {
		tmp = (t_0 * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (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))
    if (l <= (-2.5d+150)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= 8.8d-275) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + (h * (((m * (d_1 * (0.5d0 / d))) ** 2.0d0) * ((-0.5d0) / l)))))
    else
        tmp = (t_0 * (1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))))) * (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));
	double tmp;
	if (l <= -2.5e+150) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= 8.8e-275) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + (h * (Math.pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))));
	} else {
		tmp = (t_0 * (1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (Math.sqrt(d) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -2.5e+150:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= 8.8e-275:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + (h * (math.pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))))
	else:
		tmp = (t_0 * (1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (math.sqrt(d) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.5e+150)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= 8.8e-275)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(h * Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) * Float64(-0.5 / l))))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))) * Float64(sqrt(d) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -2.5e+150)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= 8.8e-275)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * (((M * (D * (0.5 / d))) ^ 2.0) * (-0.5 / l)))));
	else
		tmp = (t_0 * (1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5)))) * (sqrt(d) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.5e+150], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 8.8e-275], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(h * N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.50000000000000004e150

   1. Initial program 53.0%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in h around 0 50.4%

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

      1. frac-2neg 53.3%

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

      2. sqrt-div 69.6%

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

   6. Applied egg-rr 66.6%

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

   if -2.50000000000000004e150 < l < 8.79999999999999955e-275

   1. Initial program 76.5%

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

   3. Simplified 75.6%

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

      1. expm1-log1p-u 24.4%

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

      2. expm1-udef 24.4%

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

   5. Applied egg-rr 24.4%

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

      1. expm1-def 24.4%

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

      2. expm1-log1p 75.6%

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

      3. associate-*l/ 76.6%

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

      4. *-commutative 76.6%

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

      5. associate-*l/ 77.6%

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

      6. *-commutative 77.6%

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

      7. *-commutative 77.6%

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

      8. associate-/l* 77.6%

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

      9. associate-*r/ 79.6%

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

      10. *-commutative 79.6%

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

      11. associate-*r/ 79.6%

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

      12. associate-*r/ 79.6%

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

   7. Simplified 79.6%

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

      1. expm1-log1p-u 75.3%

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

      2. expm1-udef 68.1%

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

      3. associate-*r* 67.1%

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

      4. div-inv 67.1%

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

      5. metadata-eval 67.1%

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

   9. Applied egg-rr 67.1%

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

       1. expm1-def 74.3%

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

       2. expm1-log1p 79.5%

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

       3. *-rgt-identity 79.5%

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

       4. associate-*r/ 79.5%

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

       5. *-commutative 79.5%

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

       6. associate-*r* 77.6%

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

       7. *-commutative 77.6%

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

       8. associate-/r* 77.6%

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

       9. metadata-eval 77.6%

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

   11. Simplified 77.6%

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

   if 8.79999999999999955e-275 < l

   1. Initial program 72.3%

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

   3. Simplified 73.8%

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

      1. sqrt-div 84.4%

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

      2. div-inv 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \]

Alternative 3: 74.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))))
    (/ (sqrt d) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (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) :: tmp
    if (l <= (-2d-310)) then
        tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / 2.0d0) * (m / d)) ** 2.0d0))))
    else
        tmp = (sqrt((d / l)) * (1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))))) * (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 tmp;
	if (l <= -2e-310) {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (Math.sqrt(d) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = (math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * (math.sqrt(d) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))) * Float64(sqrt(d) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5)))) * (sqrt(d) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 70.7%

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

   3. Simplified 69.9%

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

      1. frac-2neg 69.9%

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

      2. sqrt-div 73.7%

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

   5. Applied egg-rr 73.7%

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

   if -1.999999999999994e-310 < l

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. sqrt-div 84.4%

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

      2. div-inv 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 79.2%

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

Alternative 4: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))))))
   (if (<= l -2e-310)
     (* (/ (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 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5)));
	double tmp;
	if (l <= -2e-310) {
		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 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))))
    if (l <= (-2d-310)) 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 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)));
	double tmp;
	if (l <= -2e-310) {
		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 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))
	tmp = 0
	if l <= -2e-310:
		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(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))))
	tmp = 0.0
	if (l <= -2e-310)
		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 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5)));
	tmp = 0.0;
	if (l <= -2e-310)
		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[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], 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 l < -1.999999999999994e-310

   1. Initial program 70.7%

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

   3. Simplified 69.9%

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

      1. frac-2neg 69.9%

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

      2. sqrt-div 82.5%

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

   5. Applied egg-rr 82.5%

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

   if -1.999999999999994e-310 < l

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. sqrt-div 84.4%

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

      2. div-inv 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 83.5%

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

Alternative 5: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -5.5e+152)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (*
      (sqrt (/ d h))
      (* t_0 (+ 1.0 (* h (* (pow (* M (* D (/ 0.5 d))) 2.0) (/ -0.5 l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -5.5e+152) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * (pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-5.5d+152)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + (h * (((m * (d_1 * (0.5d0 / d))) ** 2.0d0) * ((-0.5d0) / l)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -5.5e+152) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + (h * (Math.pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -5.5e+152:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	else:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + (h * (math.pow((M * (D * (0.5 / d))), 2.0) * (-0.5 / l)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5.5e+152)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(h * Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) * Float64(-0.5 / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -5.5e+152)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	else
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * (((M * (D * (0.5 / d))) ^ 2.0) * (-0.5 / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.5e+152], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(h * N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.4999999999999999e152

   1. Initial program 53.0%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in h around 0 50.4%

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

      1. frac-2neg 53.3%

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

      2. sqrt-div 69.6%

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

   6. Applied egg-rr 66.6%

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

   if -5.4999999999999999e152 < l

   1. Initial program 74.2%

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

   3. Simplified 74.6%

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

      1. expm1-log1p-u 29.9%

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

      2. expm1-udef 29.9%

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

   5. Applied egg-rr 29.9%

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

      1. expm1-def 29.9%

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

      2. expm1-log1p 74.6%

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

      3. associate-*l/ 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-*l/ 76.0%

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

      6. *-commutative 76.0%

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

      7. *-commutative 76.0%

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

      8. associate-/l* 76.0%

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

      9. associate-*r/ 76.0%

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

      10. *-commutative 76.0%

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

      11. associate-*r/ 76.4%

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

      12. associate-*r/ 76.4%

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

   7. Simplified 76.4%

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

      1. expm1-log1p-u 56.8%

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

      2. expm1-udef 51.0%

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

      3. associate-*r* 50.6%

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

      4. div-inv 50.6%

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

      5. metadata-eval 50.6%

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

   9. Applied egg-rr 50.6%

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

       1. expm1-def 55.9%

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

       2. expm1-log1p 76.0%

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

       3. *-rgt-identity 76.0%

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

       4. associate-*r/ 76.0%

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

       5. *-commutative 76.0%

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

       6. associate-*r* 76.0%

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

       7. *-commutative 76.0%

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

       8. associate-/r* 76.0%

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

       9. metadata-eval 76.0%

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

   11. Simplified 76.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 6: 56.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.4e+79)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (fma (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5) 1.0))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e+79) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * fma((h / l), (pow(((M / 2.0) * (D / d)), 2.0) * -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.4e+79)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(Float64(h / l), Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.4e+79], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.40000000000000005e79

   1. Initial program 60.2%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in h around 0 46.6%

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

      1. frac-2neg 60.4%

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

      2. sqrt-div 78.0%

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

   6. Applied egg-rr 64.2%

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

   if -6.40000000000000005e79 < l

   1. Initial program 74.5%

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

   3. Simplified 75.0%

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

      1. sqrt-div 54.2%

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

      2. div-inv 54.2%

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

   5. Applied egg-rr 54.2%

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

      1. associate-*r/ 54.2%

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

      2. *-rgt-identity 54.2%

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

   7. Simplified 54.2%

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

      1. expm1-log1p-u 27.1%

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

      2. expm1-udef 19.4%

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

   9. Applied egg-rr 19.6%

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

       1. expm1-def 26.6%

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

       2. expm1-log1p 66.4%

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

       3. unpow1/2 66.4%

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

       4. times-frac 66.8%

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

   11. Simplified 66.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5, 1\right)\\ \end{array} \]

Alternative 7: 56.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.1e+67)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M d) (* D 0.5)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e+67) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * pow(((M / d) * (D * 0.5)), 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 (l <= (-1.1d+67)) then
        tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h / l) * (0.5d0 * (((m / d) * (d_1 * 0.5d0)) ** 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 (l <= -1.1e+67) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M / d) * (D * 0.5)), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.1e+67:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * math.pow(((M / d) * (D * 0.5)), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.1e+67)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.1e+67)
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * (((M / d) * (D * 0.5)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.1e+67], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1e67

   1. Initial program 60.2%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in h around 0 46.6%

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

      1. frac-2neg 60.4%

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

      2. sqrt-div 78.0%

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

   6. Applied egg-rr 64.2%

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

   if -1.1e67 < l

   1. Initial program 74.5%

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

   3. Simplified 75.0%

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

      1. sqrt-div 54.2%

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

      2. div-inv 54.2%

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

   5. Applied egg-rr 52.7%

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

      1. expm1-log1p-u 27.1%

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

      2. expm1-udef 19.4%

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

   7. Applied egg-rr 19.6%

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

      1. expm1-def 26.6%

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

      2. expm1-log1p 66.9%

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

      3. unpow1/2 66.9%

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

      4. *-commutative 66.9%

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

   9. Simplified 66.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 46.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 1.95e+62)
   (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M d) (* D 0.5)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 1.95e+62) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * pow(((M / d) * (D * 0.5)), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.95d+62) then
        tmp = sqrt((d / l)) * (1.0d0 / sqrt((h / d)))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h / l) * (0.5d0 * (((m / d) * (d_1 * 0.5d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 1.95e+62) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M / d) * (D * 0.5)), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 1.95e+62:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * math.pow(((M / d) * (D * 0.5)), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 1.95e+62)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 1.95e+62)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h / l) * (0.5 * (((M / d) * (D * 0.5)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 1.95e+62], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.95e62

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in h around 0 39.0%

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

      1. clear-num 38.6%

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

      2. sqrt-div 39.8%

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

      3. metadata-eval 39.8%

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

   6. Applied egg-rr 39.8%

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

   if 1.95e62 < D

   1. Initial program 73.1%

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

   3. Simplified 73.2%

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

      1. sqrt-div 45.6%

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

      2. div-inv 45.5%

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

   5. Applied egg-rr 43.2%

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

      1. expm1-log1p-u 10.9%

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

      2. expm1-udef 8.8%

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

   7. Applied egg-rr 7.2%

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

      1. expm1-def 11.5%

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

      2. expm1-log1p 64.3%

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

      3. unpow1/2 64.3%

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

      4. *-commutative 64.3%

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

   9. Simplified 64.3%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 42.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{{\left(\left(-d\right) \cdot \frac{d}{h}\right)}^{0.5}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.05e+74)
   (/ (pow (* (- d) (/ d h)) 0.5) (sqrt (- l)))
   (if (<= h -5.5e-291)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (* d (pow (* l h) -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.05e+74) {
		tmp = pow((-d * (d / h)), 0.5) / sqrt(-l);
	} else if (h <= -5.5e-291) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1.05d+74)) then
        tmp = ((-d * (d / h)) ** 0.5d0) / sqrt(-l)
    else if (h <= (-5.5d-291)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.05e+74) {
		tmp = Math.pow((-d * (d / h)), 0.5) / Math.sqrt(-l);
	} else if (h <= -5.5e-291) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.05e+74:
		tmp = math.pow((-d * (d / h)), 0.5) / math.sqrt(-l)
	elif h <= -5.5e-291:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.05e+74)
		tmp = Float64((Float64(Float64(-d) * Float64(d / h)) ^ 0.5) / sqrt(Float64(-l)));
	elseif (h <= -5.5e-291)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.05e+74)
		tmp = ((-d * (d / h)) ^ 0.5) / sqrt(-l);
	elseif (h <= -5.5e-291)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.05e+74], N[(N[Power[N[((-d) * N[(d / h), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5.5e-291], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.0499999999999999e74

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in h around 0 30.3%

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

      1. add-log-exp 8.1%

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

      2. *-rgt-identity 8.1%

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

      3. *-commutative 8.1%

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

      4. pow1/2 8.1%

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

      5. pow1/2 8.1%

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

      6. pow-prod-down 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. rem-log-exp 27.7%

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

      2. *-commutative 27.7%

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

      3. unpow-prod-down 30.3%

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

      4. pow1/2 30.3%

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

      5. pow1/2 30.3%

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

      6. frac-2neg 30.3%

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

      7. sqrt-undiv 34.4%

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

      8. associate-*r/ 34.5%

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

      9. pow1/2 34.5%

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

      10. pow1/2 34.5%

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

      11. pow-prod-down 32.3%

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

   8. Applied egg-rr 32.3%

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

   if -1.0499999999999999e74 < h < -5.5000000000000002e-291

   1. Initial program 70.4%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in h around 0 32.9%

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

      1. add-log-exp 13.6%

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

      2. *-rgt-identity 13.6%

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

      3. *-commutative 13.6%

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

      4. pow1/2 13.6%

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

      5. pow1/2 13.6%

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

      6. pow-prod-down 13.6%

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

   6. Applied egg-rr 13.6%

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

   7. Taylor expanded in d around -inf 50.4%

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

      1. associate-*r* 50.4%

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

      2. neg-mul-1 50.4%

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

   9. Simplified 50.4%

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

   if -5.5000000000000002e-291 < h

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in h around 0 37.4%

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

   5. Taylor expanded in d around 0 43.6%

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

      1. *-commutative 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in d around 0 43.6%

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

      1. associate-/l/ 43.6%

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

      2. unpow1/2 43.6%

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

      3. associate-/l/ 43.6%

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

      4. rem-exp-log 41.6%

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

      5. exp-neg 41.6%

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

      6. exp-prod 41.6%

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

      7. distribute-lft-neg-out 41.6%

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

      8. distribute-rgt-neg-in 41.6%

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

      9. metadata-eval 41.6%

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

      10. exp-to-pow 43.6%

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

   10. Simplified 43.6%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{{\left(\left(-d\right) \cdot \frac{d}{h}\right)}^{0.5}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 10: 41.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-134}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9.2e-134)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.2e-134) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-9.2d-134)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.2e-134) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -9.2e-134:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9.2e-134)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -9.2e-134)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9.2e-134], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.2000000000000001e-134

   1. Initial program 67.1%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in h around 0 35.1%

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

      1. add-log-exp 10.4%

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

      2. *-rgt-identity 10.4%

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

      3. *-commutative 10.4%

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

      4. pow1/2 10.4%

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

      5. pow1/2 10.4%

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

      6. pow-prod-down 10.4%

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

   6. Applied egg-rr 10.4%

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

   7. Taylor expanded in d around -inf 42.1%

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

      1. associate-*r* 42.1%

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

      2. neg-mul-1 42.1%

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

   9. Simplified 42.1%

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

   if -9.2000000000000001e-134 < l

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in h around 0 34.6%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in d around 0 41.7%

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

      1. associate-/l/ 41.7%

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

      2. unpow1/2 41.7%

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

      3. associate-/l/ 41.7%

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

      4. rem-exp-log 40.1%

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

      5. exp-neg 40.1%

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

      6. exp-prod 40.1%

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

      7. distribute-lft-neg-out 40.1%

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

      8. distribute-rgt-neg-in 40.1%

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

      9. metadata-eval 40.1%

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

      10. exp-to-pow 41.7%

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

   10. Simplified 41.7%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-134}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 11: 36.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3.6e-134) (sqrt (* (/ d l) (/ d h))) (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.6e-134) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-3.6d-134)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.6e-134) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -3.6e-134:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3.6e-134)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -3.6e-134)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3.6e-134], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.5999999999999999e-134

   1. Initial program 67.1%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in h around 0 35.1%

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

      1. expm1-log1p-u 34.4%

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

      2. expm1-udef 21.1%

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

      3. *-rgt-identity 21.1%

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

      4. *-commutative 21.1%

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

      5. pow1/2 21.1%

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

      6. pow1/2 21.1%

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

      7. pow-prod-down 20.1%

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

   6. Applied egg-rr 20.1%

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

      1. expm1-def 30.1%

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

      2. expm1-log1p 30.8%

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

      3. unpow1/2 30.8%

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

   8. Simplified 30.8%

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

   if -3.5999999999999999e-134 < l

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in h around 0 34.6%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in d around 0 41.7%

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

      1. associate-/l/ 41.7%

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

      2. unpow1/2 41.7%

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

      3. associate-/l/ 41.7%

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

      4. rem-exp-log 40.1%

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

      5. exp-neg 40.1%

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

      6. exp-prod 40.1%

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

      7. distribute-lft-neg-out 40.1%

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

      8. distribute-rgt-neg-in 40.1%

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

      9. metadata-eval 40.1%

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

      10. exp-to-pow 41.7%

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

   10. Simplified 41.7%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 12: 41.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-132}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -1.75e-132) (* d (- t_0)) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -1.75e-132) {
		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 = (l * h) ** (-0.5d0)
    if (l <= (-1.75d-132)) 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((l * h), -0.5);
	double tmp;
	if (l <= -1.75e-132) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -1.75e-132:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -1.75e-132)
		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 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -1.75e-132)
		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[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -1.75e-132], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.75e-132

   1. Initial program 67.1%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in h around 0 35.1%

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

      1. add-log-exp 10.4%

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

      2. *-rgt-identity 10.4%

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

      3. *-commutative 10.4%

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

      4. pow1/2 10.4%

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

      5. pow1/2 10.4%

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

      6. pow-prod-down 10.4%

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

   6. Applied egg-rr 10.4%

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

   7. Taylor expanded in d around -inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. *-commutative 42.1%

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

      3. distribute-rgt-neg-in 42.1%

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

      4. associate-/l/ 42.1%

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

      5. unpow1/2 42.1%

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

      6. associate-/l/ 42.1%

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

      7. rem-exp-log 39.9%

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

      8. exp-neg 39.9%

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

      9. exp-prod 39.9%

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

      10. distribute-lft-neg-out 39.9%

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

      11. distribute-rgt-neg-in 39.9%

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

      12. metadata-eval 39.9%

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

      13. exp-to-pow 42.1%

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

   9. Simplified 42.1%

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

   if -1.75e-132 < l

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in h around 0 34.6%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in d around 0 41.7%

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

      1. associate-/l/ 41.7%

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

      2. unpow1/2 41.7%

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

      3. associate-/l/ 41.7%

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

      4. rem-exp-log 40.1%

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

      5. exp-neg 40.1%

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

      6. exp-prod 40.1%

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

      7. distribute-lft-neg-out 40.1%

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

      8. distribute-rgt-neg-in 40.1%

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

      9. metadata-eval 40.1%

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

      10. exp-to-pow 41.7%

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

   10. Simplified 41.7%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-132}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 13: 26.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

3. Simplified 71.8%

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

4. Taylor expanded in h around 0 34.8%

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

5. Taylor expanded in d around 0 28.8%

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

   1. *-commutative 28.8%

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

7. Simplified 28.8%

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

8. Taylor expanded in d around 0 28.8%

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

   1. associate-/l/ 28.8%

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

   2. unpow1/2 28.8%

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

   3. associate-/l/ 28.8%

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

   4. rem-exp-log 27.8%

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

   5. exp-neg 27.8%

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

   6. exp-prod 27.8%

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

   7. distribute-lft-neg-out 27.8%

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

   8. distribute-rgt-neg-in 27.8%

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

   9. metadata-eval 27.8%

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

   10. exp-to-pow 28.8%

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

10. Simplified 28.8%

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

11. Final simplification 28.8%

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

Alternative 14: 4.4% accurate, 332.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 0.0)

C

double code(double d, double h, double l, double M, double D) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return 0.0;
}

Python

def code(d, h, l, M, D):
	return 0.0

Julia

function code(d, h, l, M, D)
	return 0.0
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = 0.0;
end

Wolfram

code[d_, h_, l_, M_, D_] := 0.0

Derivation

1. Initial program 71.8%

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

3. Simplified 71.8%

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

4. Taylor expanded in h around 0 34.8%

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

   1. add-log-exp 13.0%

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

   2. *-rgt-identity 13.0%

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

   3. *-commutative 13.0%

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

   4. pow1/2 13.0%

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

   5. pow1/2 13.0%

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

   6. pow-prod-down 13.0%

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

6. Applied egg-rr 13.0%

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

7. Taylor expanded in d around 0 6.5%

   \[\leadsto \log \color{blue}{1} \]

8. Final simplification 6.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023339 
(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)))))