Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 85.4%

Time: 31.7s

Alternatives: 20

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: 66.6% 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: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{-132}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0)) (t_1 (sqrt (- d))))
   (if (<= l -4e-310)
     (*
      (/ t_1 (sqrt (- l)))
      (*
       (/ t_1 (sqrt (- h)))
       (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0))) l))))
     (if (<= l 7.4e-132)
       (*
        (* (sqrt (/ d h)) (/ (sqrt d) (sqrt l)))
        (- 1.0 (* 0.5 (* h (/ t_0 l)))))
       (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_1 / sqrt(-l)) * ((t_1 / sqrt(-h)) * (1.0 + ((h * (-0.5 * pow((D * (M / (d * 2.0))), 2.0))) / l)));
	} else if (l <= 7.4e-132) {
		tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * (h * (t_0 / l))));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0))) / l))));
	elseif (l <= 7.4e-132)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.4e-132], 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[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

      1. frac-2neg 63.1%

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

      2. sqrt-div 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. associate-*l/ 78.1%

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

      2. *-commutative 78.1%

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

      3. add-sqr-sqrt 78.1%

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

      4. pow2 78.1%

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

      5. sqrt-pow1 78.1%

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

      6. metadata-eval 78.1%

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

      7. pow1 78.1%

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

      8. associate-/l/ 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. frac-2neg 78.1%

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

      2. sqrt-div 94.0%

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

   10. Applied egg-rr 94.0%

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

   if -3.999999999999988e-310 < l < 7.4000000000000004e-132

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 7.4000000000000004e-132 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{-132}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t\_2}{\sqrt{-\ell}} \cdot \left(t\_0 \cdot \left(1 + \frac{{\left(\frac{M \cdot \frac{D}{2}}{d} \cdot \sqrt{h \cdot -0.5}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\ell \leq -3.25 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-\mathsf{fma}\left(\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-130}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{t\_1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_1, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (pow (/ (* D M) (* d 2.0)) 2.0))
        (t_2 (sqrt (- d)))
        (t_3
         (*
          (/ t_2 (sqrt (- l)))
          (*
           t_0
           (+
            1.0
            (/ (pow (* (/ (* M (/ D 2.0)) d) (sqrt (* h -0.5))) 2.0) l))))))
   (if (<= l -1.3e+64)
     t_3
     (if (<= l -3.25e-155)
       (*
        (/ t_2 (sqrt (- h)))
        (*
         (sqrt (/ d l))
         (* h (- (fma (/ (pow (* D (/ M d)) 2.0) l) 0.125 (/ -1.0 h))))))
       (if (<= l -4e-310)
         t_3
         (if (<= l 1.95e-130)
           (* (* t_0 (/ (sqrt d) (sqrt l))) (- 1.0 (* 0.5 (* h (/ t_1 l)))))
           (* d (/ (fma (* -0.5 (/ h l)) t_1 1.0) (* (sqrt l) (sqrt h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_2 = sqrt(-d);
	double t_3 = (t_2 / sqrt(-l)) * (t_0 * (1.0 + (pow((((M * (D / 2.0)) / d) * sqrt((h * -0.5))), 2.0) / l)));
	double tmp;
	if (l <= -1.3e+64) {
		tmp = t_3;
	} else if (l <= -3.25e-155) {
		tmp = (t_2 / sqrt(-h)) * (sqrt((d / l)) * (h * -fma((pow((D * (M / d)), 2.0) / l), 0.125, (-1.0 / h))));
	} else if (l <= -4e-310) {
		tmp = t_3;
	} else if (l <= 1.95e-130) {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * (h * (t_1 / l))));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_1, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(Float64(M * Float64(D / 2.0)) / d) * sqrt(Float64(h * -0.5))) ^ 2.0) / l))))
	tmp = 0.0
	if (l <= -1.3e+64)
		tmp = t_3;
	elseif (l <= -3.25e-155)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(h * Float64(-fma(Float64((Float64(D * Float64(M / d)) ^ 2.0) / l), 0.125, Float64(-1.0 / h))))));
	elseif (l <= -4e-310)
		tmp = t_3;
	elseif (l <= 1.95e-130)
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_1 / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_1, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(N[(M * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e+64], t$95$3, If[LessEqual[l, -3.25e-155], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(h * (-N[(N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 0.125 + N[(-1.0 / h), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], t$95$3, If[LessEqual[l, 1.95e-130], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.29999999999999998e64 or -3.25e-155 < l < -3.999999999999988e-310

   1. Initial program 59.1%

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

   3. Simplified 59.0%

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

      1. frac-2neg 59.0%

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

      2. sqrt-div 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. associate-*l/ 80.9%

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

      2. *-commutative 80.9%

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

      3. add-sqr-sqrt 80.9%

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

      4. pow2 80.9%

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

      5. sqrt-pow1 80.9%

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

      6. metadata-eval 80.9%

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

      7. pow1 80.9%

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

      8. associate-/l/ 80.9%

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

   8. Applied egg-rr 80.9%

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

      1. add-sqr-sqrt 80.9%

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

      2. *-un-lft-identity 80.9%

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

      3. times-frac 80.9%

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

      4. associate-*r* 80.9%

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

      5. sqrt-prod 81.0%

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

      6. sqrt-pow1 64.0%

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

      7. metadata-eval 64.0%

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

      8. pow1 64.0%

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

      9. associate-*r/ 64.0%

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

   10. Applied egg-rr 84.5%

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

       1. /-rgt-identity 84.5%

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

       2. associate-*r/ 84.5%

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

       3. unpow2 84.5%

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

       4. *-commutative 84.5%

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

       5. *-commutative 84.5%

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

       6. associate-/r* 84.5%

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

       7. associate-*l/ 84.5%

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

   12. Simplified 84.5%

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

   if -1.29999999999999998e64 < l < -3.25e-155

   1. Initial program 71.3%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in h around -inf 42.0%

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

      1. mul-1-neg 42.0%

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

      2. distribute-rgt-neg-in 42.0%

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

      3. *-commutative 42.0%

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

      4. fma-neg 42.0%

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

   7. Simplified 72.2%

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

      1. frac-2neg 72.2%

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

      2. sqrt-div 96.9%

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

   9. Applied egg-rr 97.1%

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

   if -3.999999999999988e-310 < l < 1.95e-130

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 1.95e-130 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{{\left(\frac{M \cdot \frac{D}{2}}{d} \cdot \sqrt{h \cdot -0.5}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -3.25 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-\mathsf{fma}\left(\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 0.125, \frac{-1}{h}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{{\left(\frac{M \cdot \frac{D}{2}}{d} \cdot \sqrt{h \cdot -0.5}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-\ell}}\\ t_3 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-123}:\\ \;\;\;\;t\_2 \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \left(1 + \frac{{\left(\frac{M \cdot \frac{D}{2}}{d} \cdot \sqrt{h \cdot -0.5}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.35 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{t\_3}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_3, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (- d)))
        (t_2 (/ t_1 (sqrt (- l))))
        (t_3 (pow (/ (* D M) (* d 2.0)) 2.0)))
   (if (<= l -6.5e-123)
     (*
      t_2
      (*
       (/ t_1 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D (/ (/ M 2.0) d)) 2.0))))))
     (if (<= l -4e-310)
       (*
        t_2
        (*
         t_0
         (+ 1.0 (/ (pow (* (/ (* M (/ D 2.0)) d) (sqrt (* h -0.5))) 2.0) l))))
       (if (<= l 4.35e-131)
         (* (* t_0 (/ (sqrt d) (sqrt l))) (- 1.0 (* 0.5 (* h (/ t_3 l)))))
         (* d (/ (fma (* -0.5 (/ h l)) t_3 1.0) (* (sqrt l) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt(-d);
	double t_2 = t_1 / sqrt(-l);
	double t_3 = pow(((D * M) / (d * 2.0)), 2.0);
	double tmp;
	if (l <= -6.5e-123) {
		tmp = t_2 * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D * ((M / 2.0) / d)), 2.0)))));
	} else if (l <= -4e-310) {
		tmp = t_2 * (t_0 * (1.0 + (pow((((M * (D / 2.0)) / d) * sqrt((h * -0.5))), 2.0) / l)));
	} else if (l <= 4.35e-131) {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * (h * (t_3 / l))));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_3, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(t_1 / sqrt(Float64(-l)))
	t_3 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	tmp = 0.0
	if (l <= -6.5e-123)
		tmp = Float64(t_2 * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0))))));
	elseif (l <= -4e-310)
		tmp = Float64(t_2 * Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(Float64(M * Float64(D / 2.0)) / d) * sqrt(Float64(h * -0.5))) ^ 2.0) / l))));
	elseif (l <= 4.35e-131)
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_3 / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_3, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -6.5e-123], N[(t$95$2 * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(t$95$2 * N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(N[(M * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.35e-131], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.49999999999999938e-123

   1. Initial program 56.1%

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

   3. Simplified 56.0%

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

      1. frac-2neg 56.0%

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

      2. sqrt-div 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. frac-2neg 70.7%

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

      2. sqrt-div 91.9%

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

   8. Applied egg-rr 85.4%

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

   if -6.49999999999999938e-123 < l < -3.999999999999988e-310

   1. Initial program 74.6%

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

   3. Simplified 74.6%

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

      1. frac-2neg 74.6%

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

      2. sqrt-div 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. associate-*l/ 90.1%

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

      2. *-commutative 90.1%

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

      3. add-sqr-sqrt 90.1%

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

      4. pow2 90.1%

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

      5. sqrt-pow1 90.1%

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

      6. metadata-eval 90.1%

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

      7. pow1 90.1%

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

      8. associate-/l/ 90.1%

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

   8. Applied egg-rr 90.1%

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

      1. add-sqr-sqrt 90.1%

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

      2. *-un-lft-identity 90.1%

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

      3. times-frac 90.1%

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

      4. associate-*r* 90.1%

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

      5. sqrt-prod 90.1%

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

      6. sqrt-pow1 71.0%

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

      7. metadata-eval 71.0%

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

      8. pow1 71.0%

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

      9. associate-*r/ 71.0%

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

   10. Applied egg-rr 92.3%

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

       1. /-rgt-identity 92.3%

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

       2. associate-*r/ 92.2%

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

       3. unpow2 92.2%

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

       4. *-commutative 92.2%

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

       5. *-commutative 92.2%

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

       6. associate-/r* 92.2%

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

       7. associate-*l/ 92.2%

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

   12. Simplified 92.2%

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

   if -3.999999999999988e-310 < l < 4.3499999999999998e-131

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 4.3499999999999998e-131 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{{\left(\frac{M \cdot \frac{D}{2}}{d} \cdot \sqrt{h \cdot -0.5}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.35 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \frac{t\_2}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot t\_3\right)\\ \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+147}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, t\_0, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\ell \leq 8.3 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{t\_1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(t\_0, t\_1, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ h l)))
        (t_1 (pow (/ (* D M) (* d 2.0)) 2.0))
        (t_2 (sqrt (- d)))
        (t_3 (sqrt (/ d h)))
        (t_4
         (*
          (/ t_2 (sqrt (- l)))
          (*
           (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0))) l))
           t_3))))
   (if (<= l -5.1e+147)
     t_4
     (if (<= l -3.6e-128)
       (*
        (/ t_2 (sqrt (- h)))
        (* (sqrt (/ d l)) (fma (pow (* 0.5 (* M (/ D d))) 2.0) t_0 1.0)))
       (if (<= l -4e-310)
         t_4
         (if (<= l 8.3e-131)
           (* (* t_3 (/ (sqrt d) (sqrt l))) (- 1.0 (* 0.5 (* h (/ t_1 l)))))
           (* d (/ (fma t_0 t_1 1.0) (* (sqrt l) (sqrt h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = -0.5 * (h / l);
	double t_1 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_2 = sqrt(-d);
	double t_3 = sqrt((d / h));
	double t_4 = (t_2 / sqrt(-l)) * ((1.0 + ((h * (-0.5 * pow((D * (M / (d * 2.0))), 2.0))) / l)) * t_3);
	double tmp;
	if (l <= -5.1e+147) {
		tmp = t_4;
	} else if (l <= -3.6e-128) {
		tmp = (t_2 / sqrt(-h)) * (sqrt((d / l)) * fma(pow((0.5 * (M * (D / d))), 2.0), t_0, 1.0));
	} else if (l <= -4e-310) {
		tmp = t_4;
	} else if (l <= 8.3e-131) {
		tmp = (t_3 * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * (h * (t_1 / l))));
	} else {
		tmp = d * (fma(t_0, t_1, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(-0.5 * Float64(h / l))
	t_1 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_2 = sqrt(Float64(-d))
	t_3 = sqrt(Float64(d / h))
	t_4 = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0))) / l)) * t_3))
	tmp = 0.0
	if (l <= -5.1e+147)
		tmp = t_4;
	elseif (l <= -3.6e-128)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * fma((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0), t_0, 1.0)));
	elseif (l <= -4e-310)
		tmp = t_4;
	elseif (l <= 8.3e-131)
		tmp = Float64(Float64(t_3 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_1 / l)))));
	else
		tmp = Float64(d * Float64(fma(t_0, t_1, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.1e+147], t$95$4, If[LessEqual[l, -3.6e-128], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], t$95$4, If[LessEqual[l, 8.3e-131], N[(N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(t$95$0 * t$95$1 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.09999999999999999e147 or -3.60000000000000025e-128 < l < -3.999999999999988e-310

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

      1. frac-2neg 58.3%

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

      2. sqrt-div 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. associate-*l/ 81.1%

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

      2. *-commutative 81.1%

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

      3. add-sqr-sqrt 81.1%

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

      4. pow2 81.1%

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

      5. sqrt-pow1 81.1%

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

      6. metadata-eval 81.1%

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

      7. pow1 81.1%

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

      8. associate-/l/ 81.1%

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

   8. Applied egg-rr 81.1%

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

   if -5.09999999999999999e147 < l < -3.60000000000000025e-128

   1. Initial program 71.0%

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

   3. Simplified 68.9%

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

      1. frac-2neg 73.0%

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

      2. sqrt-div 91.2%

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

   6. Applied egg-rr 87.1%

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

   if -3.999999999999988e-310 < l < 8.29999999999999963e-131

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 8.29999999999999963e-131 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 8.3 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, h \cdot -0.125, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0)))
   (if (<= l -4e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (sqrt (/ d l)) (fma (/ (pow (* D (/ M d)) 2.0) l) (* h -0.125) 1.0)))
     (if (<= l 1.04e-131)
       (*
        (* (sqrt (/ d h)) (/ (sqrt d) (sqrt l)))
        (- 1.0 (* 0.5 (* h (/ t_0 l)))))
       (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double tmp;
	if (l <= -4e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * fma((pow((D * (M / d)), 2.0) / l), (h * -0.125), 1.0));
	} else if (l <= 1.04e-131) {
		tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * (h * (t_0 / l))));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * fma(Float64((Float64(D * Float64(M / d)) ^ 2.0) / l), Float64(h * -0.125), 1.0)));
	elseif (l <= 1.04e-131)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.04e-131], 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[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 63.1%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in M around 0 38.2%

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

      1. +-commutative 38.2%

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

      2. associate-*r/ 38.2%

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

      3. associate-*r* 38.3%

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

      4. associate-*r* 38.3%

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

      5. associate-*l/ 39.0%

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

      6. associate-*r/ 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-*l* 39.0%

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

      9. fma-define 39.0%

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

   7. Simplified 68.4%

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

      1. frac-2neg 78.1%

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

      2. sqrt-div 94.0%

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

   9. Applied egg-rr 81.1%

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

   if -3.999999999999988e-310 < l < 1.04e-131

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 1.04e-131 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, h \cdot -0.125, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_1 := 1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(t\_3 \cdot t\_2\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0))
        (t_1 (- 1.0 (* 0.5 (* h (/ t_0 l)))))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (/ d h))))
   (if (<= l -1e-124)
     (*
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D (/ (/ M 2.0) d)) 2.0)))))
      t_2)
     (if (<= l -4e-310)
       (* t_1 (* t_3 t_2))
       (if (<= l 6.5e-131)
         (* (* t_3 (/ (sqrt d) (sqrt l))) t_1)
         (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_1 = 1.0 - (0.5 * (h * (t_0 / l)));
	double t_2 = sqrt((d / l));
	double t_3 = sqrt((d / h));
	double tmp;
	if (l <= -1e-124) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D * ((M / 2.0) / d)), 2.0))))) * t_2;
	} else if (l <= -4e-310) {
		tmp = t_1 * (t_3 * t_2);
	} else if (l <= 6.5e-131) {
		tmp = (t_3 * (sqrt(d) / sqrt(l))) * t_1;
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_1 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l))))
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -1e-124)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0))))) * t_2);
	elseif (l <= -4e-310)
		tmp = Float64(t_1 * Float64(t_3 * t_2));
	elseif (l <= 6.5e-131)
		tmp = Float64(Float64(t_3 * Float64(sqrt(d) / sqrt(l))) * t_1);
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-124], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, -4e-310], N[(t$95$1 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-131], N[(N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.99999999999999933e-125

   1. Initial program 56.6%

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

   3. Simplified 56.5%

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

      1. frac-2neg 71.1%

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

      2. sqrt-div 92.0%

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

   6. Applied egg-rr 72.3%

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

   if -9.99999999999999933e-125 < l < -3.999999999999988e-310

   1. Initial program 74.1%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 77.6%

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

      2. frac-times 81.9%

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

      3. associate-/l* 77.6%

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

      4. *-commutative 77.6%

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

   6. Applied egg-rr 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/l* 77.6%

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

      3. associate-*r/ 81.9%

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

      4. *-commutative 81.9%

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

   8. Simplified 81.9%

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

   if -3.999999999999988e-310 < l < 6.5000000000000002e-131

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 6.5000000000000002e-131 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\right) \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0)) (t_1 (sqrt (- d))))
   (if (<= d -2.15e-156)
     (*
      (- 1.0 (* 0.5 (* h (/ t_0 l))))
      (* (/ t_1 (sqrt (- h))) (sqrt (/ d l))))
     (if (<= d -5e-310)
       (*
        (/ t_1 (sqrt (- l)))
        (*
         (+ 1.0 (/ (* h (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0))) l))
         (sqrt (/ d h))))
       (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -2.15e-156) {
		tmp = (1.0 - (0.5 * (h * (t_0 / l)))) * ((t_1 / sqrt(-h)) * sqrt((d / l)));
	} else if (d <= -5e-310) {
		tmp = (t_1 / sqrt(-l)) * ((1.0 + ((h * (-0.5 * pow((D * (M / (d * 2.0))), 2.0))) / l)) * sqrt((d / h)));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -2.15e-156)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l)))) * Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0))) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -2.15e-156], N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.14999999999999989e-156

   1. Initial program 71.7%

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

   3. Simplified 69.5%

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

      1. associate-*r/ 76.6%

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

      2. frac-times 78.8%

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

      3. associate-/l* 76.6%

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

      4. *-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/l* 76.6%

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

      3. associate-*r/ 78.8%

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

      4. *-commutative 78.8%

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

   8. Simplified 78.8%

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

      1. frac-2neg 83.3%

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

      2. sqrt-div 96.2%

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

   10. Applied egg-rr 89.5%

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

   if -2.14999999999999989e-156 < d < -4.999999999999985e-310

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

      1. frac-2neg 38.3%

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

      2. sqrt-div 56.4%

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

   6. Applied egg-rr 56.4%

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

      1. associate-*l/ 62.9%

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

      2. *-commutative 62.9%

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

      3. add-sqr-sqrt 62.9%

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

      4. pow2 62.9%

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

      5. sqrt-pow1 62.9%

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

      6. metadata-eval 62.9%

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

      7. pow1 62.9%

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

      8. associate-/l/ 62.9%

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

   8. Applied egg-rr 62.9%

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

   if -4.999999999999985e-310 < d

   1. Initial program 61.9%

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

   3. Simplified 60.4%

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

   6. Applied egg-rr 73.9%

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

      1. unpow1 73.9%

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

      2. associate-*l/ 75.6%

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

      3. associate-/l* 77.0%

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

      4. +-commutative 77.0%

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

      5. associate-*r* 77.0%

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

      6. fma-define 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-*r/ 77.7%

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

      9. *-commutative 77.7%

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

   8. Simplified 77.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_1 := 1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-132}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0))
        (t_1 (- 1.0 (* 0.5 (* h (/ t_0 l)))))
        (t_2 (sqrt (/ d h))))
   (if (<= l -4e-310)
     (* t_1 (* t_2 (sqrt (/ d l))))
     (if (<= l 2.5e-132)
       (* (* t_2 (/ (sqrt d) (sqrt l))) t_1)
       (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_1 = 1.0 - (0.5 * (h * (t_0 / l)));
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= -4e-310) {
		tmp = t_1 * (t_2 * sqrt((d / l)));
	} else if (l <= 2.5e-132) {
		tmp = (t_2 * (sqrt(d) / sqrt(l))) * t_1;
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_1 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l))))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(t_1 * Float64(t_2 * sqrt(Float64(d / l))));
	elseif (l <= 2.5e-132)
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(l))) * t_1);
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(t$95$1 * N[(t$95$2 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-132], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 63.1%

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

   3. Simplified 61.5%

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

      1. associate-*r/ 66.8%

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

      2. frac-times 68.4%

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

      3. associate-/l* 66.8%

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

      4. *-commutative 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/l* 66.8%

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

      3. associate-*r/ 68.4%

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

      4. *-commutative 68.4%

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

   8. Simplified 68.4%

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

   if -3.999999999999988e-310 < l < 2.5e-132

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 2.5e-132 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-132}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ t_1 := 1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0))
        (t_1 (- 1.0 (* 0.5 (* h (/ t_0 l))))))
   (if (<= l -4e-310)
     (* t_1 (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (if (<= l 9.6e-131)
       (* (* (sqrt (/ d h)) (/ (sqrt d) (sqrt l))) t_1)
       (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double t_1 = 1.0 - (0.5 * (h * (t_0 / l)));
	double tmp;
	if (l <= -4e-310) {
		tmp = t_1 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else if (l <= 9.6e-131) {
		tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * t_1;
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	t_1 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l))))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(t_1 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (l <= 9.6e-131)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) / sqrt(l))) * t_1);
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(t$95$1 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.6e-131], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 63.1%

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

   3. Simplified 61.5%

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

      1. associate-*r/ 66.8%

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

      2. frac-times 68.4%

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

      3. associate-/l* 66.8%

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

      4. *-commutative 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/l* 66.8%

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

      3. associate-*r/ 68.4%

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

      4. *-commutative 68.4%

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

   8. Simplified 68.4%

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

      1. frac-2neg 78.1%

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

      2. sqrt-div 94.0%

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

   10. Applied egg-rr 78.0%

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

   if -3.999999999999988e-310 < l < 9.5999999999999999e-131

   1. Initial program 65.6%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 73.4%

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

      2. frac-times 75.9%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 92.3%

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

   10. Applied egg-rr 92.3%

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

   if 9.5999999999999999e-131 < l

   1. Initial program 60.3%

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

   3. Simplified 59.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow1 73.3%

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

      2. associate-*l/ 75.8%

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

      3. associate-/l* 77.8%

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

      4. +-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. fma-define 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r/ 77.8%

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

      9. *-commutative 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq 6.4 \cdot 10^{-262}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{t\_0}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* D M) (* d 2.0)) 2.0)))
   (if (<= l 6.4e-262)
     (* (- 1.0 (* 0.5 (* h (/ t_0 l)))) (* (sqrt (/ d h)) (sqrt (/ d l))))
     (* d (/ (fma (* -0.5 (/ h l)) t_0 1.0) (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D * M) / (d * 2.0)), 2.0);
	double tmp;
	if (l <= 6.4e-262) {
		tmp = (1.0 - (0.5 * (h * (t_0 / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = d * (fma((-0.5 * (h / l)), t_0, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0
	tmp = 0.0
	if (l <= 6.4e-262)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 / l)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.5 * Float64(h / l)), t_0, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 6.4e-262], N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 6.4000000000000001e-262

   1. Initial program 62.5%

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

   3. Simplified 61.1%

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

      1. associate-*r/ 68.1%

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

      2. frac-times 69.6%

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

      3. associate-/l* 68.1%

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

      4. *-commutative 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-/l* 68.1%

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

      3. associate-*r/ 69.6%

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

      4. *-commutative 69.6%

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

   8. Simplified 69.6%

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

   if 6.4000000000000001e-262 < l

   1. Initial program 62.4%

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

   3. Simplified 60.8%

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

   6. Applied egg-rr 74.0%

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

      1. unpow1 74.0%

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

      2. associate-*l/ 75.9%

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

      3. associate-/l* 77.3%

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

      4. +-commutative 77.3%

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

      5. associate-*r* 77.3%

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

      6. fma-define 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-*r/ 78.2%

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

      9. *-commutative 78.2%

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

   8. Simplified 78.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{-262}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+157}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.25e+157)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -1.18e-170)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D (/ (/ M 2.0) d)) 2.0))))))
     (if (<= l 6.2e-298)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (- 1.0 (* 0.5 (* h (/ (pow (* D (/ M (* d 2.0))) 2.0) l)))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (+ 1.0 (* (* -0.5 (/ h l)) (pow (* D (/ (/ M d) 2.0)) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.25e+157) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -1.18e-170) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D * ((M / 2.0) / d)), 2.0)))));
	} else if (l <= 6.2e-298) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.25d+157)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= (-1.18d-170)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_1 * ((m / 2.0d0) / d)) ** 2.0d0)))))
    else if (l <= 6.2d-298) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) / l))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((-0.5d0) * (h / l)) * ((d_1 * ((m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.25e+157) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -1.18e-170) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D * ((M / 2.0) / d)), 2.0)))));
	} else if (l <= 6.2e-298) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (Math.pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * Math.pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.25e+157:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= -1.18e-170:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D * ((M / 2.0) / d)), 2.0)))))
	elif l <= 6.2e-298:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (math.pow((D * (M / (d * 2.0))), 2.0) / l))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * math.pow((D * ((M / d) / 2.0)), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.25e+157)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -1.18e-170)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0))))));
	elseif (l <= 6.2e-298)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.25e+157)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= -1.18e-170)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((D * ((M / 2.0) / d)) ^ 2.0)))));
	elseif (l <= 6.2e-298)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (((D * (M / (d * 2.0))) ^ 2.0) / l))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * ((D * ((M / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.25e+157], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.18e-170], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e-298], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.24999999999999992e157

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in d around inf 5.8%

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

      1. associate-/r* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 52.1%

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

      7. neg-mul-1 52.1%

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

   10. Simplified 52.1%

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

   if -2.24999999999999992e157 < l < -1.18e-170

   1. Initial program 68.4%

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

   3. Simplified 68.3%

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

   if -1.18e-170 < l < 6.2000000000000003e-298

   1. Initial program 74.7%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 83.6%

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

      2. frac-times 86.5%

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

      3. associate-/l* 83.6%

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

      4. *-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-/l* 83.6%

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

      3. associate-*r/ 86.5%

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

      4. *-commutative 86.5%

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

   8. Simplified 86.5%

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

      1. pow1 86.5%

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

      2. sqrt-unprod 86.5%

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

      3. associate-*r* 86.5%

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

      4. associate-/l* 86.5%

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

   10. Applied egg-rr 86.5%

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

       1. unpow1 86.5%

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

       2. associate-*l* 86.5%

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

       3. *-commutative 86.5%

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

   12. Simplified 86.5%

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

   if 6.2000000000000003e-298 < l

   1. Initial program 62.0%

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

   3. Simplified 60.6%

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

   6. Applied egg-rr 74.2%

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

      1. unpow1 74.2%

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

      2. associate-*r* 74.2%

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

      3. *-commutative 74.2%

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

      4. associate-*r/ 75.0%

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

      5. *-commutative 75.0%

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

      6. associate-*r/ 75.0%

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

      7. associate-*r* 75.0%

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

      8. associate-*r* 75.0%

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

      9. associate-/r* 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+157}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.25e+145)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 4.8e-296)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ M (* d 2.0))) 2.0) l)))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (* -0.5 (/ h l)) (pow (* D (/ (/ M d) 2.0)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.25e+145) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 4.8e-296) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.25d+145)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 4.8d-296) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) / l))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((-0.5d0) * (h / l)) * ((d_1 * ((m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.25e+145) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 4.8e-296) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (Math.pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * Math.pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.25e+145:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 4.8e-296:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (math.pow((D * (M / (d * 2.0))), 2.0) / l))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * math.pow((D * ((M / d) / 2.0)), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.25e+145)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 4.8e-296)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.25e+145)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 4.8e-296)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (((D * (M / (d * 2.0))) ^ 2.0) / l))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * ((D * ((M / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.25e+145], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e-296], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.24999999999999992e145

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in d around inf 5.3%

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

      1. associate-/r* 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 49.0%

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

      7. neg-mul-1 49.0%

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

   10. Simplified 49.0%

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

   if -1.24999999999999992e145 < l < 4.79999999999999992e-296

   1. Initial program 72.6%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 75.3%

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

      2. frac-times 77.4%

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

      3. associate-/l* 75.3%

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

      4. *-commutative 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 75.3%

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

      3. associate-*r/ 77.4%

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

      4. *-commutative 77.4%

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

   8. Simplified 77.4%

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

      1. pow1 77.4%

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

      2. sqrt-unprod 68.9%

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

      3. associate-*r* 68.9%

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

      4. associate-/l* 68.9%

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

   10. Applied egg-rr 68.9%

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

       1. unpow1 68.9%

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

       2. associate-*l* 68.9%

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

       3. *-commutative 68.9%

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

   12. Simplified 68.9%

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

   if 4.79999999999999992e-296 < l

   1. Initial program 62.0%

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

   3. Simplified 60.6%

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

   6. Applied egg-rr 74.2%

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

      1. unpow1 74.2%

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

      2. associate-*r* 74.2%

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

      3. *-commutative 74.2%

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

      4. associate-*r/ 75.0%

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

      5. *-commutative 75.0%

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

      6. associate-*r/ 75.0%

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

      7. associate-*r* 75.0%

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

      8. associate-*r* 75.0%

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

      9. associate-/r* 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-197}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.5e-197)
   (*
    (- 1.0 (* 0.5 (* h (/ (pow (/ (* D M) (* d 2.0)) 2.0) l))))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* (* -0.5 (/ h l)) (pow (* D (/ (/ M d) 2.0)) 2.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-197) {
		tmp = (1.0 - (0.5 * (h * (pow(((D * M) / (d * 2.0)), 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.5d-197) then
        tmp = (1.0d0 - (0.5d0 * (h * ((((d_1 * m) / (d * 2.0d0)) ** 2.0d0) / l)))) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((-0.5d0) * (h / l)) * ((d_1 * ((m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-197) {
		tmp = (1.0 - (0.5 * (h * (Math.pow(((D * M) / (d * 2.0)), 2.0) / l)))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * Math.pow((D * ((M / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.5e-197:
		tmp = (1.0 - (0.5 * (h * (math.pow(((D * M) / (d * 2.0)), 2.0) / l)))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * math.pow((D * ((M / d) / 2.0)), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.5e-197)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) / l)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.5e-197)
		tmp = (1.0 - (0.5 * (h * ((((D * M) / (d * 2.0)) ^ 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * ((D * ((M / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.5e-197], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e-197

   1. Initial program 63.0%

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

   3. Simplified 61.0%

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

      1. associate-*r/ 68.0%

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

      2. frac-times 70.0%

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

      3. associate-/l* 68.0%

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

      4. *-commutative 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-/l* 68.0%

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

      3. associate-*r/ 70.0%

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

      4. *-commutative 70.0%

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

   8. Simplified 70.0%

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

   if 2.5000000000000001e-197 < l

   1. Initial program 61.7%

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

   3. Simplified 60.9%

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

   6. Applied egg-rr 74.0%

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

      1. unpow1 74.0%

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

      2. associate-*r* 74.0%

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

      3. *-commutative 74.0%

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

      4. associate-*r/ 74.0%

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

      5. *-commutative 74.0%

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

      6. associate-*r/ 74.1%

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

      7. associate-*r* 74.1%

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

      8. associate-*r* 74.1%

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

      9. associate-/r* 74.1%

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

   8. Simplified 74.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-197}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.56 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4.5e+145)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 1.56e+53)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ M (* d 2.0))) 2.0) l)))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.5e+145) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 1.56e+53) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-4.5d+145)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 1.56d+53) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) / l))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.5e+145) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 1.56e+53) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (Math.pow((D * (M / (d * 2.0))), 2.0) / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -4.5e+145:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 1.56e+53:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (math.pow((D * (M / (d * 2.0))), 2.0) / l))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -4.5e+145)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 1.56e+53)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -4.5e+145)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 1.56e+53)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * (h * (((D * (M / (d * 2.0))) ^ 2.0) / l))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4.5e+145], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.56e+53], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4999999999999998e145

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in d around inf 5.3%

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

      1. associate-/r* 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 49.0%

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

      7. neg-mul-1 49.0%

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

   10. Simplified 49.0%

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

   if -4.4999999999999998e145 < l < 1.56e53

   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 68.6%

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

      1. associate-*r/ 73.2%

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

      2. frac-times 75.1%

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

      3. associate-/l* 73.2%

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

      4. *-commutative 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-/l* 73.2%

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

      3. associate-*r/ 75.0%

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

      4. *-commutative 75.0%

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

   8. Simplified 75.0%

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

      1. pow1 75.0%

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

      2. sqrt-unprod 66.8%

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

      3. associate-*r* 66.8%

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

      4. associate-/l* 65.6%

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

   10. Applied egg-rr 65.6%

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

       1. unpow1 65.6%

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

       2. associate-*l* 65.6%

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

       3. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   if 1.56e53 < l

   1. Initial program 55.1%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in d around inf 47.8%

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

      1. associate-/r* 47.8%

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

   7. Simplified 47.8%

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

      1. sqrt-div 59.0%

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

      2. div-inv 59.0%

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

      3. inv-pow 59.0%

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

      4. sqrt-pow1 59.0%

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

      5. metadata-eval 59.0%

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

   9. Applied egg-rr 59.0%

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

       1. associate-*r/ 59.0%

          \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5} \cdot 1}{\sqrt{\ell}}} \]

       2. *-rgt-identity 59.0%

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

   11. Simplified 59.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.56 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1e-300)
   (* (- d) (pow (* l h) -0.5))
   (* d (/ (pow h -0.5) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1e-300) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 1d-300) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1e-300) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 1e-300:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1e-300)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1e-300)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1e-300], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.00000000000000003e-300

   1. Initial program 63.4%

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

   3. Simplified 61.8%

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

      1. associate-*r/ 67.1%

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

      2. frac-times 68.7%

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

      3. associate-/l* 67.1%

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

      4. *-commutative 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-/l* 67.1%

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

      3. associate-*r/ 68.7%

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

      4. *-commutative 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow-1 0.0%

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

       3. metadata-eval 0.0%

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

       4. pow-sqr 0.0%

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

       5. rem-sqrt-square 0.0%

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

       6. rem-square-sqrt 0.0%

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

       7. fabs-sqr 0.0%

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

       8. rem-square-sqrt 0.0%

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

       9. *-commutative 0.0%

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

       10. unpow2 0.0%

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

       11. rem-square-sqrt 39.6%

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

       12. neg-mul-1 39.6%

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

   11. Simplified 39.6%

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

   if 1.00000000000000003e-300 < h

   1. Initial program 61.6%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in d around inf 43.7%

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

      1. associate-/r* 43.7%

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

   7. Simplified 43.7%

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

      1. sqrt-div 51.5%

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

      2. div-inv 51.4%

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

      3. inv-pow 51.4%

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

      4. sqrt-pow1 51.4%

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

      5. metadata-eval 51.4%

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

   9. Applied egg-rr 51.4%

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

       1. associate-*r/ 51.5%

          \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5} \cdot 1}{\sqrt{\ell}}} \]

       2. *-rgt-identity 51.5%

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

   11. Simplified 51.5%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.8e-91) (sqrt (* (/ d h) (/ d l))) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-91) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-2.8d-91)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-91) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.8e-91:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.8e-91)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.8e-91)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.8e-91], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.8e-91

   1. Initial program 73.6%

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

   3. Simplified 73.5%

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

      1. clear-num 72.6%

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

      2. sqrt-div 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in d around inf 42.5%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 36.8%

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

      3. *-commutative 36.8%

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

      4. *-commutative 36.8%

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

      5. swap-sqr 36.8%

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

      6. add-sqr-sqrt 36.9%

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

      7. frac-times 36.9%

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

      8. metadata-eval 36.9%

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

      9. add-sqr-sqrt 37.0%

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

      10. clear-num 37.0%

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

   9. Applied egg-rr 37.0%

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

   if -2.8e-91 < d

   1. Initial program 57.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in d around inf 36.9%

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

      1. associate-/r* 36.9%

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

   7. Simplified 36.9%

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

      1. pow1 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-/l/ 36.9%

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

      4. sqrt-div 36.3%

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

      5. metadata-eval 36.3%

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

   9. Applied egg-rr 36.3%

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

       1. unpow1 36.3%

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

       2. associate-*l/ 36.4%

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

       3. *-lft-identity 36.4%

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

       4. *-commutative 36.4%

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

   11. Simplified 36.4%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -7.5e-92) (sqrt (* (/ d h) (/ d l))) (* d (sqrt (/ 1.0 (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.5e-92) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-7.5d-92)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * sqrt((1.0d0 / (l * h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.5e-92) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -7.5e-92:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -7.5e-92)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -7.5e-92)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -7.5e-92], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.5000000000000005e-92

   1. Initial program 73.6%

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

   3. Simplified 73.5%

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

      1. clear-num 72.6%

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

      2. sqrt-div 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in d around inf 42.5%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 36.8%

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

      3. *-commutative 36.8%

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

      4. *-commutative 36.8%

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

      5. swap-sqr 36.8%

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

      6. add-sqr-sqrt 36.9%

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

      7. frac-times 36.9%

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

      8. metadata-eval 36.9%

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

      9. add-sqr-sqrt 37.0%

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

      10. clear-num 37.0%

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

   9. Applied egg-rr 37.0%

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

   if -7.5000000000000005e-92 < d

   1. Initial program 57.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in d around inf 36.9%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8e-92) (sqrt (* (/ d h) (/ d l))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8e-92) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-8d-92)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8e-92) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -8e-92:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8e-92)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8e-92)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8e-92], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.9999999999999999e-92

   1. Initial program 73.6%

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

   3. Simplified 73.5%

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

      1. clear-num 72.6%

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

      2. sqrt-div 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in d around inf 42.5%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 36.8%

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

      3. *-commutative 36.8%

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

      4. *-commutative 36.8%

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

      5. swap-sqr 36.8%

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

      6. add-sqr-sqrt 36.9%

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

      7. frac-times 36.9%

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

      8. metadata-eval 36.9%

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

      9. add-sqr-sqrt 37.0%

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

      10. clear-num 37.0%

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

   9. Applied egg-rr 37.0%

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

   if -7.9999999999999999e-92 < d

   1. Initial program 57.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in d around inf 36.9%

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

      1. associate-/r* 36.9%

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

   7. Simplified 36.9%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{-219}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.8e-219)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.8e-219) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-8.8d-219)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.8e-219) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.8e-219:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.8e-219)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.8e-219)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.8e-219], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.7999999999999998e-219

   1. Initial program 67.5%

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

   3. Simplified 65.6%

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

      1. associate-*r/ 71.7%

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

      2. frac-times 73.6%

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

      3. associate-/l* 71.7%

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

      4. *-commutative 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-/l* 71.7%

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

      3. associate-*r/ 73.6%

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

      4. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow-1 0.0%

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

       3. metadata-eval 0.0%

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

       4. pow-sqr 0.0%

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

       5. rem-sqrt-square 0.0%

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

       6. rem-square-sqrt 0.0%

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

       7. fabs-sqr 0.0%

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

       8. rem-square-sqrt 0.0%

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

       9. *-commutative 0.0%

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

       10. unpow2 0.0%

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

       11. rem-square-sqrt 45.1%

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

       12. neg-mul-1 45.1%

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

   11. Simplified 45.1%

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

   if -8.7999999999999998e-219 < d

   1. Initial program 58.9%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in d around inf 41.3%

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

      1. associate-/r* 41.3%

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

   7. Simplified 41.3%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{-219}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

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

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Simplified 60.9%

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

5. Taylor expanded in d around inf 27.0%

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

   1. associate-/r* 27.0%

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

7. Simplified 27.0%

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

   1. pow1 27.0%

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

   2. *-commutative 27.0%

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

   3. associate-/l/ 27.0%

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

   4. sqrt-div 26.6%

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

   5. metadata-eval 26.6%

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

9. Applied egg-rr 26.6%

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

    1. unpow1 26.6%

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

    2. associate-*l/ 26.7%

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

    3. *-lft-identity 26.7%

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

    4. *-commutative 26.7%

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

11. Simplified 26.7%

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

12. Final simplification 26.7%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))))