Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 82.0%

Time: 29.4s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% 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: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -3 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -3e+142)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (/ (* D (* M 0.5)) d) 2.0) -0.5)))))
     (if (<= h -5e-310)
       (*
        (* (/ 1.0 (sqrt (/ h d))) (/ t_0 (sqrt (- l))))
        (- 1.0 (* 0.5 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l))))
       (*
        d
        (/
         (fma -0.5 (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)) 1.0)
         (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -3e+142) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D * (M * 0.5)) / d), 2.0) * -0.5))));
	} else if (h <= -5e-310) {
		tmp = ((1.0 / sqrt((h / d))) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else {
		tmp = d * (fma(-0.5, (h * (pow((D * (M * (0.5 / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -3e+142)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0) * -0.5)))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64(fma(-0.5, Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -3e+142], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.99999999999999975e142

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

      1. frac-2neg 56.2%

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

      2. sqrt-div 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. associate-*r/ 73.3%

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

      2. div-inv 73.3%

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

      3. metadata-eval 73.3%

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

   8. Applied egg-rr 73.3%

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

   if -2.99999999999999975e142 < h < -4.999999999999985e-310

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

      1. associate-*r/ 75.8%

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

      2. *-commutative 75.8%

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

      3. div-inv 75.8%

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

      4. metadata-eval 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 87.6%

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

   8. Applied egg-rr 87.6%

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

      1. clear-num 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

   10. Applied egg-rr 87.6%

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

   if -4.999999999999985e-310 < h

   1. Initial program 66.0%

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

   3. Simplified 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-rgt-in 58.3%

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

      3. *-un-lft-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. sqrt-div 62.0%

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

      6. sqrt-div 62.6%

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

      7. frac-times 62.5%

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

      8. add-sqr-sqrt 62.6%

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

   6. Applied egg-rr 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. distribute-lft-in 81.4%

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

      4. associate-*l/ 83.2%

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

      5. associate-/l* 83.8%

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

   8. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -3 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -3e+143)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (/ (* D (* M 0.5)) d) 2.0) -0.5)))))
     (if (<= h -5e-310)
       (*
        (- 1.0 (* 0.5 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l)))
        (* (/ t_0 (sqrt (- l))) (sqrt (/ d h))))
       (*
        d
        (/
         (fma -0.5 (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)) 1.0)
         (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -3e+143) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D * (M * 0.5)) / d), 2.0) * -0.5))));
	} else if (h <= -5e-310) {
		tmp = (1.0 - (0.5 * ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l))) * ((t_0 / sqrt(-l)) * sqrt((d / h)));
	} else {
		tmp = d * (fma(-0.5, (h * (pow((D * (M * (0.5 / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -3e+143)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0) * -0.5)))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))) * Float64(Float64(t_0 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * Float64(fma(-0.5, Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -3e+143], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.0000000000000001e143

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

      1. frac-2neg 56.2%

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

      2. sqrt-div 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. associate-*r/ 73.3%

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

      2. div-inv 73.3%

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

      3. metadata-eval 73.3%

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

   8. Applied egg-rr 73.3%

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

   if -3.0000000000000001e143 < h < -4.999999999999985e-310

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

      1. associate-*r/ 75.8%

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

      2. *-commutative 75.8%

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

      3. div-inv 75.8%

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

      4. metadata-eval 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 87.6%

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

   8. Applied egg-rr 87.6%

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

   if -4.999999999999985e-310 < h

   1. Initial program 66.0%

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

   3. Simplified 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-rgt-in 58.3%

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

      3. *-un-lft-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. sqrt-div 62.0%

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

      6. sqrt-div 62.6%

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

      7. frac-times 62.5%

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

      8. add-sqr-sqrt 62.6%

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

   6. Applied egg-rr 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. distribute-lft-in 81.4%

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

      4. associate-*l/ 83.2%

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

      5. associate-/l* 83.8%

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

   8. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 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 < -1.999999999999994e-310

   1. Initial program 68.3%

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

   3. Simplified 66.6%

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

      1. frac-2neg 66.6%

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

      2. sqrt-div 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. associate-*r/ 75.0%

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

      2. div-inv 75.0%

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

      3. metadata-eval 75.0%

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

   8. Applied egg-rr 75.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.0%

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

   3. Simplified 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-rgt-in 58.3%

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

      3. *-un-lft-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. sqrt-div 62.0%

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

      6. sqrt-div 62.6%

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

      7. frac-times 62.5%

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

      8. add-sqr-sqrt 62.6%

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

   6. Applied egg-rr 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. distribute-lft-in 81.4%

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

      4. associate-*l/ 83.2%

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

      5. associate-/l* 83.8%

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

   8. Simplified 86.5%

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

Alternative 4: 82.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * 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]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 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 < -1.999999999999994e-310

   1. Initial program 68.3%

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

   3. Simplified 66.6%

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

      1. frac-2neg 66.6%

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

      2. sqrt-div 73.4%

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

   6. Applied egg-rr 73.4%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.0%

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

   3. Simplified 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-rgt-in 58.3%

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

      3. *-un-lft-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. sqrt-div 62.0%

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

      6. sqrt-div 62.6%

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

      7. frac-times 62.5%

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

      8. add-sqr-sqrt 62.6%

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

   6. Applied egg-rr 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. distribute-lft-in 81.4%

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

      4. associate-*l/ 83.2%

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

      5. associate-/l* 83.8%

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

   8. Simplified 86.5%

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

4. Final simplification 80.5%

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

Alternative 5: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.1e-185)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ 1.0 l) (* h (* -0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))))
   (if (<= d -2e-310)
     (* (sqrt (* h (pow l -3.0))) (* (/ (pow (* D M) 2.0) d) 0.125))
     (*
      d
      (/
       (fma -0.5 (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.1e-185) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((1.0 / l) * (h * (-0.5 * pow(((D * M) / (d * 2.0)), 2.0))))));
	} else if (d <= -2e-310) {
		tmp = sqrt((h * pow(l, -3.0))) * ((pow((D * M), 2.0) / d) * 0.125);
	} else {
		tmp = d * (fma(-0.5, (h * (pow((D * (M * (0.5 / d))), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.1e-185)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(1.0 / l) * Float64(h * Float64(-0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))))));
	elseif (d <= -2e-310)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(Float64((Float64(D * M) ^ 2.0) / d) * 0.125));
	else
		tmp = Float64(d * Float64(fma(-0.5, Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.1e-185], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[(-0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(-0.5 * N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 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 < -2.1e-185

   1. Initial program 78.5%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 80.7%

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

      2. clear-num 80.7%

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

      3. *-commutative 80.7%

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

      4. add-sqr-sqrt 80.7%

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

      5. pow2 80.7%

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

      6. unpow2 80.7%

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

      7. sqrt-prod 49.3%

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

      8. add-sqr-sqrt 80.7%

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

      9. associate-/l/ 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. associate-/r/ 80.7%

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

      2. associate-*r/ 83.0%

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

   8. Simplified 83.0%

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

   if -2.1e-185 < d < -1.999999999999994e-310

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 53.0%

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

      7. *-commutative 53.0%

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

      8. mul-1-neg 53.0%

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

   7. Simplified 53.0%

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

      1. associate-*r/ 53.0%

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

      2. div-inv 53.0%

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

      3. pow-flip 53.0%

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

      4. metadata-eval 53.0%

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

      5. pow-prod-down 53.3%

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

   9. Applied egg-rr 53.3%

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

       1. associate-/l* 53.1%

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

       2. distribute-rgt-neg-out 53.1%

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

       3. distribute-lft-neg-in 53.1%

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

       4. metadata-eval 53.1%

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

       5. associate-*r/ 53.1%

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

       6. unpow2 53.1%

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

       7. swap-sqr 53.0%

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

       8. unpow2 53.0%

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

       9. unpow2 53.0%

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

       10. *-commutative 53.0%

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

       11. unpow2 53.0%

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

       12. unpow2 53.0%

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

       13. swap-sqr 53.1%

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

       14. unpow2 53.1%

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

   11. Simplified 53.1%

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

   if -1.999999999999994e-310 < d

   1. Initial program 66.0%

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

   3. Simplified 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-rgt-in 58.3%

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

      3. *-un-lft-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. sqrt-div 62.0%

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

      6. sqrt-div 62.6%

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

      7. frac-times 62.5%

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

      8. add-sqr-sqrt 62.6%

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

   6. Applied egg-rr 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. distribute-lft-in 81.4%

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

      4. associate-*l/ 83.2%

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

      5. associate-/l* 83.8%

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

   8. Simplified 86.5%

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

Alternative 6: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(D \cdot M\right)}^{2}}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{-269}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(t\_0 \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(t\_0 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (pow (* D M) 2.0) d))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= d -5.2e-269)
     (* t_1 (* t_2 (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D d)) 2.0))))))
     (if (<= d 3.2e-299)
       (* (sqrt (* h (pow l -3.0))) (* t_0 0.125))
       (if (<= d 5.1e-133)
         (* (sqrt (/ h (pow l 3.0))) (* t_0 -0.125))
         (*
          t_2
          (*
           t_1
           (+ 1.0 (* -0.5 (* (/ h l) (pow (* (* M 0.5) (/ D d)) 2.0)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0) / d;
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (d <= -5.2e-269) {
		tmp = t_1 * (t_2 * (1.0 + (-0.125 * ((h / l) * pow((M * (D / d)), 2.0)))));
	} else if (d <= 3.2e-299) {
		tmp = sqrt((h * pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 5.1e-133) {
		tmp = sqrt((h / pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = t_2 * (t_1 * (1.0 + (-0.5 * ((h / l) * pow(((M * 0.5) * (D / d)), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((d_1 * m) ** 2.0d0) / d
    t_1 = sqrt((d / l))
    t_2 = sqrt((d / h))
    if (d <= (-5.2d-269)) then
        tmp = t_1 * (t_2 * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_1 / d)) ** 2.0d0)))))
    else if (d <= 3.2d-299) then
        tmp = sqrt((h * (l ** (-3.0d0)))) * (t_0 * 0.125d0)
    else if (d <= 5.1d-133) then
        tmp = sqrt((h / (l ** 3.0d0))) * (t_0 * (-0.125d0))
    else
        tmp = t_2 * (t_1 * (1.0d0 + ((-0.5d0) * ((h / l) * (((m * 0.5d0) * (d_1 / d)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0) / d;
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (d <= -5.2e-269) {
		tmp = t_1 * (t_2 * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D / d)), 2.0)))));
	} else if (d <= 3.2e-299) {
		tmp = Math.sqrt((h * Math.pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 5.1e-133) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = t_2 * (t_1 * (1.0 + (-0.5 * ((h / l) * Math.pow(((M * 0.5) * (D / d)), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0) / d
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if d <= -5.2e-269:
		tmp = t_1 * (t_2 * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D / d)), 2.0)))))
	elif d <= 3.2e-299:
		tmp = math.sqrt((h * math.pow(l, -3.0))) * (t_0 * 0.125)
	elif d <= 5.1e-133:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (t_0 * -0.125)
	else:
		tmp = t_2 * (t_1 * (1.0 + (-0.5 * ((h / l) * math.pow(((M * 0.5) * (D / d)), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(D * M) ^ 2.0) / d)
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -5.2e-269)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D / d)) ^ 2.0))))));
	elseif (d <= 3.2e-299)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(t_0 * 0.125));
	elseif (d <= 5.1e-133)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(t_0 * -0.125));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D * M) ^ 2.0) / d;
	t_1 = sqrt((d / l));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (d <= -5.2e-269)
		tmp = t_1 * (t_2 * (1.0 + (-0.125 * ((h / l) * ((M * (D / d)) ^ 2.0)))));
	elseif (d <= 3.2e-299)
		tmp = sqrt((h * (l ^ -3.0))) * (t_0 * 0.125);
	elseif (d <= 5.1e-133)
		tmp = sqrt((h / (l ^ 3.0))) * (t_0 * -0.125);
	else
		tmp = t_2 * (t_1 * (1.0 + (-0.5 * ((h / l) * (((M * 0.5) * (D / d)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.2e-269], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-299], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.1e-133], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.2e-269

   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 71.8%

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

      1. associate-*l/ 74.6%

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

      2. clear-num 74.6%

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

      3. *-commutative 74.6%

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

      4. add-sqr-sqrt 74.6%

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

      5. pow2 74.6%

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

      6. unpow2 74.6%

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

      7. sqrt-prod 45.7%

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

      8. add-sqr-sqrt 74.6%

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

      9. associate-/l/ 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. associate-/r/ 74.6%

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

      2. associate-*r/ 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in l around 0 44.1%

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

       1. associate-*r* 46.1%

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

       2. times-frac 44.4%

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

       3. associate-*l/ 45.2%

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

       4. unpow2 45.2%

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

       5. unpow2 45.2%

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

       6. times-frac 55.7%

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

       7. unpow2 55.7%

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

       8. swap-sqr 73.6%

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

       9. unpow2 73.6%

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

       10. *-commutative 73.6%

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

       11. *-commutative 73.6%

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

   11. Simplified 73.6%

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

   if -5.2e-269 < d < 3.20000000000000008e-299

   1. Initial program 25.6%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 55.2%

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

      7. *-commutative 55.2%

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

      8. mul-1-neg 55.2%

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

   7. Simplified 55.2%

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

      1. associate-*r/ 55.2%

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

      2. div-inv 55.2%

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

      3. pow-flip 55.2%

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

      4. metadata-eval 55.2%

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

      5. pow-prod-down 55.4%

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

   9. Applied egg-rr 55.4%

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

       1. associate-/l* 55.2%

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

       2. distribute-rgt-neg-out 55.2%

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

       3. distribute-lft-neg-in 55.2%

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

       4. metadata-eval 55.2%

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

       5. associate-*r/ 55.2%

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

       6. unpow2 55.2%

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

       7. swap-sqr 55.2%

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

       8. unpow2 55.2%

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

       9. unpow2 55.2%

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

       10. *-commutative 55.2%

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

       11. unpow2 55.2%

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

       12. unpow2 55.2%

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

       13. swap-sqr 55.2%

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

       14. unpow2 55.2%

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

   11. Simplified 55.2%

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

   if 3.20000000000000008e-299 < d < 5.0999999999999999e-133

   1. Initial program 40.4%

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

   3. Simplified 40.4%

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

      1. add-sqr-sqrt 40.4%

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

      2. pow2 40.4%

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

      3. sqrt-prod 40.4%

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

      4. sqrt-pow1 49.4%

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

      5. metadata-eval 49.4%

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

      6. pow1 49.4%

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

      7. *-commutative 49.4%

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

      8. div-inv 49.4%

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

      9. metadata-eval 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in d around 0 46.3%

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

      1. associate-*r* 46.3%

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

      2. *-commutative 46.3%

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

      3. unpow2 46.3%

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

      4. unpow2 46.3%

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

      5. swap-sqr 68.1%

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

      6. unpow2 68.1%

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

   9. Simplified 68.1%

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

   if 5.0999999999999999e-133 < d

   1. Initial program 75.6%

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

   3. Simplified 75.7%

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

      1. fma-undefine 75.7%

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

   6. Applied egg-rr 75.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(D \cdot M\right)}^{2}}{d}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(t\_0 \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(t\_0 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (pow (* D M) 2.0) d))
        (t_1
         (*
          (sqrt (/ d l))
          (*
           (sqrt (/ d h))
           (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D d)) 2.0))))))))
   (if (<= d -3.9e-271)
     t_1
     (if (<= d 3.2e-299)
       (* (sqrt (* h (pow l -3.0))) (* t_0 0.125))
       (if (<= d 5.5e-134) (* (sqrt (/ h (pow l 3.0))) (* t_0 -0.125)) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0) / d;
	double t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * pow((M * (D / d)), 2.0)))));
	double tmp;
	if (d <= -3.9e-271) {
		tmp = t_1;
	} else if (d <= 3.2e-299) {
		tmp = sqrt((h * pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 5.5e-134) {
		tmp = sqrt((h / pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 * m) ** 2.0d0) / d
    t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_1 / d)) ** 2.0d0)))))
    if (d <= (-3.9d-271)) then
        tmp = t_1
    else if (d <= 3.2d-299) then
        tmp = sqrt((h * (l ** (-3.0d0)))) * (t_0 * 0.125d0)
    else if (d <= 5.5d-134) then
        tmp = sqrt((h / (l ** 3.0d0))) * (t_0 * (-0.125d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0) / d;
	double t_1 = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D / d)), 2.0)))));
	double tmp;
	if (d <= -3.9e-271) {
		tmp = t_1;
	} else if (d <= 3.2e-299) {
		tmp = Math.sqrt((h * Math.pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 5.5e-134) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0) / d
	t_1 = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D / d)), 2.0)))))
	tmp = 0
	if d <= -3.9e-271:
		tmp = t_1
	elif d <= 3.2e-299:
		tmp = math.sqrt((h * math.pow(l, -3.0))) * (t_0 * 0.125)
	elif d <= 5.5e-134:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (t_0 * -0.125)
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(D * M) ^ 2.0) / d)
	t_1 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D / d)) ^ 2.0))))))
	tmp = 0.0
	if (d <= -3.9e-271)
		tmp = t_1;
	elseif (d <= 3.2e-299)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(t_0 * 0.125));
	elseif (d <= 5.5e-134)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(t_0 * -0.125));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D * M) ^ 2.0) / d;
	t_1 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * ((M * (D / d)) ^ 2.0)))));
	tmp = 0.0;
	if (d <= -3.9e-271)
		tmp = t_1;
	elseif (d <= 3.2e-299)
		tmp = sqrt((h * (l ^ -3.0))) * (t_0 * 0.125);
	elseif (d <= 5.5e-134)
		tmp = sqrt((h / (l ^ 3.0))) * (t_0 * -0.125);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.9e-271], t$95$1, If[LessEqual[d, 3.2e-299], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e-134], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.89999999999999997e-271 or 5.5000000000000002e-134 < d

   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 73.7%

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

      1. associate-*l/ 75.1%

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

      2. clear-num 75.1%

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

      3. *-commutative 75.1%

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

      4. add-sqr-sqrt 75.1%

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

      5. pow2 75.1%

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

      6. unpow2 75.1%

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

      7. sqrt-prod 45.7%

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

      8. add-sqr-sqrt 75.1%

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

      9. associate-/l/ 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r/ 75.1%

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

      2. associate-*r/ 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in l around 0 44.1%

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

       1. associate-*r* 46.0%

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

       2. times-frac 46.1%

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

       3. associate-*l/ 46.6%

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

       4. unpow2 46.6%

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

       5. unpow2 46.6%

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

       6. times-frac 59.2%

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

       7. unpow2 59.2%

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

       8. swap-sqr 74.6%

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

       9. unpow2 74.6%

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

       10. *-commutative 74.6%

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

       11. *-commutative 74.6%

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

   11. Simplified 74.6%

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

   if -3.89999999999999997e-271 < d < 3.20000000000000008e-299

   1. Initial program 25.6%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 55.2%

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

      7. *-commutative 55.2%

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

      8. mul-1-neg 55.2%

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

   7. Simplified 55.2%

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

      1. associate-*r/ 55.2%

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

      2. div-inv 55.2%

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

      3. pow-flip 55.2%

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

      4. metadata-eval 55.2%

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

      5. pow-prod-down 55.4%

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

   9. Applied egg-rr 55.4%

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

       1. associate-/l* 55.2%

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

       2. distribute-rgt-neg-out 55.2%

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

       3. distribute-lft-neg-in 55.2%

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

       4. metadata-eval 55.2%

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

       5. associate-*r/ 55.2%

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

       6. unpow2 55.2%

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

       7. swap-sqr 55.2%

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

       8. unpow2 55.2%

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

       9. unpow2 55.2%

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

       10. *-commutative 55.2%

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

       11. unpow2 55.2%

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

       12. unpow2 55.2%

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

       13. swap-sqr 55.2%

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

       14. unpow2 55.2%

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

   11. Simplified 55.2%

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

   if 3.20000000000000008e-299 < d < 5.5000000000000002e-134

   1. Initial program 40.4%

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

   3. Simplified 40.4%

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

      1. add-sqr-sqrt 40.4%

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

      2. pow2 40.4%

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

      3. sqrt-prod 40.4%

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

      4. sqrt-pow1 49.4%

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

      5. metadata-eval 49.4%

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

      6. pow1 49.4%

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

      7. *-commutative 49.4%

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

      8. div-inv 49.4%

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

      9. metadata-eval 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in d around 0 46.3%

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

      1. associate-*r* 46.3%

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

      2. *-commutative 46.3%

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

      3. unpow2 46.3%

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

      4. unpow2 46.3%

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

      5. swap-sqr 68.1%

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

      6. unpow2 68.1%

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

   9. Simplified 68.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.5e+244)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ 1.0 l) (* h (* -0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.5e+244) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((1.0 / l) * (h * (-0.5 * pow(((D * M) / (d * 2.0)), 2.0))))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.5d+244) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((1.0d0 / l) * (h * ((-0.5d0) * (((d_1 * m) / (d * 2.0d0)) ** 2.0d0))))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.5e+244) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((1.0 / l) * (h * (-0.5 * Math.pow(((D * M) / (d * 2.0)), 2.0))))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.5e+244:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((1.0 / l) * (h * (-0.5 * math.pow(((D * M) / (d * 2.0)), 2.0))))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.5e+244)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(1.0 / l) * Float64(h * Float64(-0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.5e+244)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((1.0 / l) * (h * (-0.5 * (((D * M) / (d * 2.0)) ^ 2.0))))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.5e+244], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[(-0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4999999999999999e244

   1. Initial program 70.1%

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

   3. Simplified 68.9%

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

      1. associate-*l/ 70.1%

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

      2. clear-num 70.1%

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

      3. *-commutative 70.1%

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

      4. add-sqr-sqrt 70.1%

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

      5. pow2 70.1%

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

      6. unpow2 70.1%

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

      7. sqrt-prod 41.3%

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

      8. add-sqr-sqrt 70.1%

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

      9. associate-/l/ 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. associate-/r/ 70.1%

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

      2. associate-*r/ 71.3%

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

   8. Simplified 71.3%

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

   if 1.4999999999999999e244 < l

   1. Initial program 24.5%

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

   3. Simplified 30.4%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. sqrt-prod 30.4%

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

      4. sqrt-pow1 46.8%

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

      5. metadata-eval 46.8%

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

      6. pow1 46.8%

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

      7. *-commutative 46.8%

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

      8. div-inv 46.8%

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

      9. metadata-eval 46.8%

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in d around inf 65.3%

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

      1. unpow-1 65.3%

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

      2. metadata-eval 65.3%

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

      3. pow-sqr 65.4%

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

      4. rem-sqrt-square 65.4%

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

      5. rem-square-sqrt 65.3%

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

      6. fabs-sqr 65.3%

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

      7. rem-square-sqrt 65.4%

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

   9. Simplified 65.4%

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

       1. *-commutative 65.4%

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

       2. unpow-prod-down 70.8%

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

   11. Applied egg-rr 70.8%

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

Alternative 9: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+113}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.6e+113)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.6e+113) {
		tmp = (1.0 - (0.5 * ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.6d+113) then
        tmp = (1.0d0 - (0.5d0 * ((h * (((m * 0.5d0) * (d_1 / d)) ** 2.0d0)) / l))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.6e+113) {
		tmp = (1.0 - (0.5 * ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.6e+113:
		tmp = (1.0 - (0.5 * ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.6e+113)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.6e+113)
		tmp = (1.0 - (0.5 * ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.6e+113], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5999999999999999e113

   1. Initial program 71.4%

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

   3. Simplified 70.9%

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

      1. associate-*r/ 72.8%

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

      2. *-commutative 72.8%

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

      3. div-inv 72.8%

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

      4. metadata-eval 72.8%

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

   6. Applied egg-rr 72.8%

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

   if 1.5999999999999999e113 < l

   1. Initial program 44.5%

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

   3. Simplified 44.6%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

      3. sqrt-prod 44.6%

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

      4. sqrt-pow1 53.7%

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

      5. metadata-eval 53.7%

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

      6. pow1 53.7%

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

      7. *-commutative 53.7%

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

      8. div-inv 53.7%

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

      9. metadata-eval 53.7%

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

   6. Applied egg-rr 53.7%

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

   7. Taylor expanded in d around inf 52.4%

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

      1. unpow-1 52.4%

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

      2. metadata-eval 52.4%

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

      3. pow-sqr 52.4%

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

      4. rem-sqrt-square 52.4%

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

      5. rem-square-sqrt 52.2%

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

      6. fabs-sqr 52.2%

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

      7. rem-square-sqrt 52.4%

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

   9. Simplified 52.4%

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

       1. *-commutative 52.4%

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

       2. unpow-prod-down 61.2%

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

   11. Applied egg-rr 61.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+113}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(D \cdot M\right)}^{2}}{d}\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(t\_0 \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(t\_0 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (pow (* D M) 2.0) d)) (t_1 (sqrt (* (/ d l) (/ d h)))))
   (if (<= d -6e-75)
     (* (+ 1.0 (* -0.5 (* (* (/ h l) (pow (/ (* D M) d) 2.0)) 0.25))) t_1)
     (if (<= d 3.2e-299)
       (* (sqrt (* h (pow l -3.0))) (* t_0 0.125))
       (if (<= d 3.1e-103)
         (* (sqrt (/ h (pow l 3.0))) (* t_0 -0.125))
         (*
          (+ 1.0 (* -0.5 (* (pow (* M (/ D d)) 2.0) (* (/ h l) 0.25))))
          t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0) / d;
	double t_1 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (d <= -6e-75) {
		tmp = (1.0 + (-0.5 * (((h / l) * pow(((D * M) / d), 2.0)) * 0.25))) * t_1;
	} else if (d <= 3.2e-299) {
		tmp = sqrt((h * pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 3.1e-103) {
		tmp = sqrt((h / pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = (1.0 + (-0.5 * (pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 * m) ** 2.0d0) / d
    t_1 = sqrt(((d / l) * (d / h)))
    if (d <= (-6d-75)) then
        tmp = (1.0d0 + ((-0.5d0) * (((h / l) * (((d_1 * m) / d) ** 2.0d0)) * 0.25d0))) * t_1
    else if (d <= 3.2d-299) then
        tmp = sqrt((h * (l ** (-3.0d0)))) * (t_0 * 0.125d0)
    else if (d <= 3.1d-103) then
        tmp = sqrt((h / (l ** 3.0d0))) * (t_0 * (-0.125d0))
    else
        tmp = (1.0d0 + ((-0.5d0) * (((m * (d_1 / d)) ** 2.0d0) * ((h / l) * 0.25d0)))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0) / d;
	double t_1 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (d <= -6e-75) {
		tmp = (1.0 + (-0.5 * (((h / l) * Math.pow(((D * M) / d), 2.0)) * 0.25))) * t_1;
	} else if (d <= 3.2e-299) {
		tmp = Math.sqrt((h * Math.pow(l, -3.0))) * (t_0 * 0.125);
	} else if (d <= 3.1e-103) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (t_0 * -0.125);
	} else {
		tmp = (1.0 + (-0.5 * (Math.pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0) / d
	t_1 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if d <= -6e-75:
		tmp = (1.0 + (-0.5 * (((h / l) * math.pow(((D * M) / d), 2.0)) * 0.25))) * t_1
	elif d <= 3.2e-299:
		tmp = math.sqrt((h * math.pow(l, -3.0))) * (t_0 * 0.125)
	elif d <= 3.1e-103:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (t_0 * -0.125)
	else:
		tmp = (1.0 + (-0.5 * (math.pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(D * M) ^ 2.0) / d)
	t_1 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (d <= -6e-75)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(Float64(h / l) * (Float64(Float64(D * M) / d) ^ 2.0)) * 0.25))) * t_1);
	elseif (d <= 3.2e-299)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(t_0 * 0.125));
	elseif (d <= 3.1e-103)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(t_0 * -0.125));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64((Float64(M * Float64(D / d)) ^ 2.0) * Float64(Float64(h / l) * 0.25)))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D * M) ^ 2.0) / d;
	t_1 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (d <= -6e-75)
		tmp = (1.0 + (-0.5 * (((h / l) * (((D * M) / d) ^ 2.0)) * 0.25))) * t_1;
	elseif (d <= 3.2e-299)
		tmp = sqrt((h * (l ^ -3.0))) * (t_0 * 0.125);
	elseif (d <= 3.1e-103)
		tmp = sqrt((h / (l ^ 3.0))) * (t_0 * -0.125);
	else
		tmp = (1.0 + (-0.5 * (((M * (D / d)) ^ 2.0) * ((h / l) * 0.25)))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6e-75], N[(N[(1.0 + N[(-0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.2e-299], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.1e-103], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.9999999999999997e-75

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

      1. add-sqr-sqrt 80.1%

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

      2. pow2 80.1%

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

      3. sqrt-prod 80.1%

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

      4. sqrt-pow1 82.6%

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

      5. metadata-eval 82.6%

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

      6. pow1 82.6%

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

      7. *-commutative 82.6%

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

      8. div-inv 82.6%

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

      9. metadata-eval 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. add-sqr-sqrt 82.6%

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

      2. pow2 82.6%

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

   8. Applied egg-rr 82.6%

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

   10. Applied egg-rr 67.8%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 67.8%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)} \]

       2. associate-*r* 67.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot M\right)}^{2}\right) \cdot 0.25\right)}\right) \]

       3. associate-*l/ 67.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right) \cdot 0.25\right)\right) \]

   12. Simplified 67.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right)} \]

   if -5.9999999999999997e-75 < d < 3.20000000000000008e-299

   1. Initial program 46.6%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 47.3%

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

      7. *-commutative 47.3%

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

      8. mul-1-neg 47.3%

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

   7. Simplified 47.3%

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

      1. associate-*r/ 47.3%

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

      2. div-inv 47.3%

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

      3. pow-flip 49.4%

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

      4. metadata-eval 49.4%

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

      5. pow-prod-down 53.8%

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

   9. Applied egg-rr 53.8%

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

       1. associate-/l* 53.7%

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

       2. distribute-rgt-neg-out 53.7%

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

       3. distribute-lft-neg-in 53.7%

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

       4. metadata-eval 53.7%

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

       5. associate-*r/ 53.7%

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

       6. unpow2 53.7%

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

       7. swap-sqr 49.4%

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

       8. unpow2 49.4%

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

       9. unpow2 49.4%

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

       10. *-commutative 49.4%

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

       11. unpow2 49.4%

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

       12. unpow2 49.4%

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

       13. swap-sqr 53.7%

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

       14. unpow2 53.7%

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

   11. Simplified 53.7%

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

   if 3.20000000000000008e-299 < d < 3.1000000000000001e-103

   1. Initial program 45.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.8%

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

      2. pow2 45.8%

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

      3. sqrt-prod 45.7%

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

      4. sqrt-pow1 53.5%

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

      5. metadata-eval 53.5%

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

      6. pow1 53.5%

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

      7. *-commutative 53.5%

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

      8. div-inv 53.5%

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

      9. metadata-eval 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in d around 0 50.8%

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

      1. associate-*r* 50.8%

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

      2. *-commutative 50.8%

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

      3. unpow2 50.8%

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

      4. unpow2 50.8%

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

      5. swap-sqr 69.7%

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

      6. unpow2 69.7%

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

   9. Simplified 69.7%

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

   if 3.1000000000000001e-103 < d

   1. Initial program 75.3%

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

   3. Simplified 75.4%

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

      1. add-sqr-sqrt 75.4%

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

      2. pow2 75.4%

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

      3. sqrt-prod 75.4%

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

      4. sqrt-pow1 78.2%

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

      5. metadata-eval 78.2%

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

      6. pow1 78.2%

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

      7. *-commutative 78.2%

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

      8. div-inv 78.2%

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

      9. metadata-eval 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. add-sqr-sqrt 78.1%

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

      2. pow2 78.1%

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

      3. pow1/2 78.1%

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

      4. sqrt-pow1 78.1%

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

      5. metadata-eval 78.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{{\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 68.5%

          \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       2. cancel-sign-sub-inv 68.5%

          \[\leadsto \color{blue}{\left(1 + \left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       3. metadata-eval 68.5%

          \[\leadsto \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       4. *-commutative 68.5%

          \[\leadsto \left(1 + \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right) \cdot -0.5}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       5. *-commutative 68.5%

          \[\leadsto \left(1 + \color{blue}{\left(\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right) \cdot \frac{h}{\ell}\right)} \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       6. associate-*l* 68.5%

          \[\leadsto \left(1 + \color{blue}{\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)} \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       7. *-commutative 68.5%

          \[\leadsto \left(1 + \left({\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       8. *-commutative 68.5%

          \[\leadsto \left(1 + \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   12. Simplified 68.5%

       \[\leadsto \color{blue}{\left(1 + \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+58}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -28:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+112}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -1.22e+58)
     (* (+ 1.0 (* -0.5 (* (* (/ h l) (pow (/ (* D M) d) 2.0)) 0.25))) t_0)
     (if (<= l -28.0)
       (* (sqrt (* h (pow l -3.0))) (* (/ (pow (* D M) 2.0) d) 0.125))
       (if (<= l 9.8e+112)
         (* (+ 1.0 (* -0.5 (* (pow (* M (/ D d)) 2.0) (* (/ h l) 0.25)))) t_0)
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -1.22e+58) {
		tmp = (1.0 + (-0.5 * (((h / l) * pow(((D * M) / d), 2.0)) * 0.25))) * t_0;
	} else if (l <= -28.0) {
		tmp = sqrt((h * pow(l, -3.0))) * ((pow((D * M), 2.0) / d) * 0.125);
	} else if (l <= 9.8e+112) {
		tmp = (1.0 + (-0.5 * (pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_0;
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-1.22d+58)) then
        tmp = (1.0d0 + ((-0.5d0) * (((h / l) * (((d_1 * m) / d) ** 2.0d0)) * 0.25d0))) * t_0
    else if (l <= (-28.0d0)) then
        tmp = sqrt((h * (l ** (-3.0d0)))) * ((((d_1 * m) ** 2.0d0) / d) * 0.125d0)
    else if (l <= 9.8d+112) then
        tmp = (1.0d0 + ((-0.5d0) * (((m * (d_1 / d)) ** 2.0d0) * ((h / l) * 0.25d0)))) * t_0
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -1.22e+58) {
		tmp = (1.0 + (-0.5 * (((h / l) * Math.pow(((D * M) / d), 2.0)) * 0.25))) * t_0;
	} else if (l <= -28.0) {
		tmp = Math.sqrt((h * Math.pow(l, -3.0))) * ((Math.pow((D * M), 2.0) / d) * 0.125);
	} else if (l <= 9.8e+112) {
		tmp = (1.0 + (-0.5 * (Math.pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -1.22e+58:
		tmp = (1.0 + (-0.5 * (((h / l) * math.pow(((D * M) / d), 2.0)) * 0.25))) * t_0
	elif l <= -28.0:
		tmp = math.sqrt((h * math.pow(l, -3.0))) * ((math.pow((D * M), 2.0) / d) * 0.125)
	elif l <= 9.8e+112:
		tmp = (1.0 + (-0.5 * (math.pow((M * (D / d)), 2.0) * ((h / l) * 0.25)))) * t_0
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -1.22e+58)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(Float64(h / l) * (Float64(Float64(D * M) / d) ^ 2.0)) * 0.25))) * t_0);
	elseif (l <= -28.0)
		tmp = Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64(Float64((Float64(D * M) ^ 2.0) / d) * 0.125));
	elseif (l <= 9.8e+112)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64((Float64(M * Float64(D / d)) ^ 2.0) * Float64(Float64(h / l) * 0.25)))) * t_0);
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -1.22e+58)
		tmp = (1.0 + (-0.5 * (((h / l) * (((D * M) / d) ^ 2.0)) * 0.25))) * t_0;
	elseif (l <= -28.0)
		tmp = sqrt((h * (l ^ -3.0))) * ((((D * M) ^ 2.0) / d) * 0.125);
	elseif (l <= 9.8e+112)
		tmp = (1.0 + (-0.5 * (((M * (D / d)) ^ 2.0) * ((h / l) * 0.25)))) * t_0;
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.22e+58], N[(N[(1.0 + N[(-0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -28.0], N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.8e+112], N[(N[(1.0 + N[(-0.5 * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.21999999999999995e58

   1. Initial program 58.5%

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

   3. Simplified 58.4%

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

      1. add-sqr-sqrt 58.4%

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

      2. pow2 58.4%

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

      3. sqrt-prod 58.4%

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

      4. sqrt-pow1 62.5%

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

      5. metadata-eval 62.5%

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

      6. pow1 62.5%

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

      7. *-commutative 62.5%

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

      8. div-inv 62.5%

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

      9. metadata-eval 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. add-sqr-sqrt 62.5%

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

      2. pow2 62.5%

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

   8. Applied egg-rr 62.5%

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

   10. Applied egg-rr 44.9%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 44.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)} \]

       2. associate-*r* 44.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot M\right)}^{2}\right) \cdot 0.25\right)}\right) \]

       3. associate-*l/ 45.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right) \cdot 0.25\right)\right) \]

   12. Simplified 45.0%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right)} \]

   if -1.21999999999999995e58 < l < -28

   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%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 45.6%

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

      7. *-commutative 45.6%

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

      8. mul-1-neg 45.6%

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

   7. Simplified 45.6%

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

      1. associate-*r/ 45.6%

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

      2. div-inv 45.6%

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

      3. pow-flip 45.6%

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

      4. metadata-eval 45.6%

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

      5. pow-prod-down 73.1%

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

   9. Applied egg-rr 73.1%

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

       1. associate-/l* 73.1%

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

       2. distribute-rgt-neg-out 73.1%

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

       3. distribute-lft-neg-in 73.1%

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

       4. metadata-eval 73.1%

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

       5. associate-*r/ 73.1%

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

       6. unpow2 73.1%

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

       7. swap-sqr 45.6%

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

       8. unpow2 45.6%

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

       9. unpow2 45.6%

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

       10. *-commutative 45.6%

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

       11. unpow2 45.6%

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

       12. unpow2 45.6%

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

       13. swap-sqr 73.1%

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

       14. unpow2 73.1%

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

   11. Simplified 73.1%

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

   if -28 < l < 9.80000000000000008e112

   1. Initial program 76.3%

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

   3. Simplified 75.7%

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

      1. add-sqr-sqrt 75.7%

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

      2. pow2 75.7%

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

      3. sqrt-prod 75.7%

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

      4. sqrt-pow1 76.9%

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

      5. metadata-eval 76.9%

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

      6. pow1 76.9%

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

      7. *-commutative 76.9%

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

      8. div-inv 76.9%

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

      9. metadata-eval 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. add-sqr-sqrt 76.7%

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

      2. pow2 76.7%

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

      3. pow1/2 76.7%

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

      4. sqrt-pow1 76.8%

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

      5. metadata-eval 76.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 76.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. Applied egg-rr 69.5%

       \[\leadsto \color{blue}{{\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 69.5%

          \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       2. cancel-sign-sub-inv 69.5%

          \[\leadsto \color{blue}{\left(1 + \left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       3. metadata-eval 69.5%

          \[\leadsto \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       4. *-commutative 69.5%

          \[\leadsto \left(1 + \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right) \cdot -0.5}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       5. *-commutative 69.5%

          \[\leadsto \left(1 + \color{blue}{\left(\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right) \cdot \frac{h}{\ell}\right)} \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       6. associate-*l* 69.5%

          \[\leadsto \left(1 + \color{blue}{\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)} \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       7. *-commutative 69.5%

          \[\leadsto \left(1 + \left({\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       8. *-commutative 69.5%

          \[\leadsto \left(1 + \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   12. Simplified 69.5%

       \[\leadsto \color{blue}{\left(1 + \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if 9.80000000000000008e112 < l

   1. Initial program 44.5%

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

   3. Simplified 44.6%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

      3. sqrt-prod 44.6%

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

      4. sqrt-pow1 53.7%

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

      5. metadata-eval 53.7%

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

      6. pow1 53.7%

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

      7. *-commutative 53.7%

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

      8. div-inv 53.7%

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

      9. metadata-eval 53.7%

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

   6. Applied egg-rr 53.7%

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

   7. Taylor expanded in d around inf 52.4%

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

      1. unpow-1 52.4%

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

      2. metadata-eval 52.4%

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

      3. pow-sqr 52.4%

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

      4. rem-sqrt-square 52.4%

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

      5. rem-square-sqrt 52.2%

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

      6. fabs-sqr 52.2%

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

      7. rem-square-sqrt 52.4%

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

   9. Simplified 52.4%

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

       1. *-commutative 52.4%

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

       2. unpow-prod-down 61.2%

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

   11. Applied egg-rr 61.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+58}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -28:\\ \;\;\;\;\sqrt{h \cdot {\ell}^{-3}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot 0.125\right)\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+112}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{+113}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.42e+113)
   (*
    (+ 1.0 (* -0.5 (* (* (/ h l) (pow (/ (* D M) d) 2.0)) 0.25)))
    (sqrt (* (/ d l) (/ d h))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.42e+113) {
		tmp = (1.0 + (-0.5 * (((h / l) * pow(((D * M) / d), 2.0)) * 0.25))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.42d+113) then
        tmp = (1.0d0 + ((-0.5d0) * (((h / l) * (((d_1 * m) / d) ** 2.0d0)) * 0.25d0))) * sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.42e+113) {
		tmp = (1.0 + (-0.5 * (((h / l) * Math.pow(((D * M) / d), 2.0)) * 0.25))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.42e+113:
		tmp = (1.0 + (-0.5 * (((h / l) * math.pow(((D * M) / d), 2.0)) * 0.25))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.42e+113)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(Float64(h / l) * (Float64(Float64(D * M) / d) ^ 2.0)) * 0.25))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.42e+113)
		tmp = (1.0 + (-0.5 * (((h / l) * (((D * M) / d) ^ 2.0)) * 0.25))) * sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.42e+113], N[(N[(1.0 + N[(-0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.42e113

   1. Initial program 71.4%

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

   3. Simplified 70.9%

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

      1. add-sqr-sqrt 70.9%

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

      2. pow2 70.9%

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

      3. sqrt-prod 70.9%

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

      4. sqrt-pow1 72.6%

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

      5. metadata-eval 72.6%

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

      6. pow1 72.6%

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

      7. *-commutative 72.6%

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

      8. div-inv 72.6%

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

      9. metadata-eval 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. add-sqr-sqrt 72.6%

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

      2. pow2 72.6%

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

   8. Applied egg-rr 72.6%

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

   10. Applied egg-rr 62.2%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 62.2%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25\right)\right)\right)} \]

       2. associate-*r* 62.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot M\right)}^{2}\right) \cdot 0.25\right)}\right) \]

       3. associate-*l/ 62.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right) \cdot 0.25\right)\right) \]

   12. Simplified 62.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right)} \]

   if 1.42e113 < l

   1. Initial program 44.5%

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

   3. Simplified 44.6%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

      3. sqrt-prod 44.6%

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

      4. sqrt-pow1 53.7%

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

      5. metadata-eval 53.7%

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

      6. pow1 53.7%

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

      7. *-commutative 53.7%

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

      8. div-inv 53.7%

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

      9. metadata-eval 53.7%

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

   6. Applied egg-rr 53.7%

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

   7. Taylor expanded in d around inf 52.4%

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

      1. unpow-1 52.4%

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

      2. metadata-eval 52.4%

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

      3. pow-sqr 52.4%

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

      4. rem-sqrt-square 52.4%

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

      5. rem-square-sqrt 52.2%

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

      6. fabs-sqr 52.2%

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

      7. rem-square-sqrt 52.4%

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

   9. Simplified 52.4%

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

       1. *-commutative 52.4%

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

       2. unpow-prod-down 61.2%

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

   11. Applied egg-rr 61.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{+113}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot 0.25\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.6 \cdot 10^{-246}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{2}\right)}^{-0.25}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -8.6e-246)
   (* d (- (pow (* h l) -0.5)))
   (if (<= l -2e-310)
     (* d (pow (pow (* h l) 2.0) -0.25))
     (if (<= l 1.85e-216)
       (* d (/ -1.0 (sqrt (* h l))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.6e-246) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * pow(pow((h * l), 2.0), -0.25);
	} else if (l <= 1.85e-216) {
		tmp = d * (-1.0 / sqrt((h * l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-8.6d-246)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (l <= (-2d-310)) then
        tmp = d * (((h * l) ** 2.0d0) ** (-0.25d0))
    else if (l <= 1.85d-216) then
        tmp = d * ((-1.0d0) / sqrt((h * l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.6e-246) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * Math.pow(Math.pow((h * l), 2.0), -0.25);
	} else if (l <= 1.85e-216) {
		tmp = d * (-1.0 / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -8.6e-246:
		tmp = d * -math.pow((h * l), -0.5)
	elif l <= -2e-310:
		tmp = d * math.pow(math.pow((h * l), 2.0), -0.25)
	elif l <= 1.85e-216:
		tmp = d * (-1.0 / math.sqrt((h * l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -8.6e-246)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= -2e-310)
		tmp = Float64(d * ((Float64(h * l) ^ 2.0) ^ -0.25));
	elseif (l <= 1.85e-216)
		tmp = Float64(d * Float64(-1.0 / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -8.6e-246)
		tmp = d * -((h * l) ^ -0.5);
	elseif (l <= -2e-310)
		tmp = d * (((h * l) ^ 2.0) ^ -0.25);
	elseif (l <= 1.85e-216)
		tmp = d * (-1.0 / sqrt((h * l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -8.6e-246], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Power[N[Power[N[(h * l), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.85e-216], N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.59999999999999985e-246

   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 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. add-sqr-sqrt 65.6%

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

      2. pow2 65.6%

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

      3. sqrt-prod 65.5%

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

      4. sqrt-pow1 67.4%

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

      5. metadata-eval 67.4%

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

      6. pow1 67.4%

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

      7. *-commutative 67.4%

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

      8. div-inv 67.4%

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

      9. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 39.8%

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

      4. neg-mul-1 39.8%

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

      5. unpow-1 39.8%

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

      6. metadata-eval 39.8%

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

      7. pow-sqr 39.8%

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

      8. rem-sqrt-square 40.2%

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

      9. rem-square-sqrt 40.0%

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

      10. fabs-sqr 40.0%

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

      11. rem-square-sqrt 40.2%

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

   9. Simplified 40.2%

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

   if -8.59999999999999985e-246 < l < -1.999999999999994e-310

   1. Initial program 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in d around inf 54.5%

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

      1. log1p-expm1-u 73.4%

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

      2. pow1/2 73.4%

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

      3. inv-pow 73.4%

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

      4. pow-pow 73.4%

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

      5. metadata-eval 73.4%

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

   7. Applied egg-rr 73.4%

      \[\leadsto d \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 54.5%

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

      2. sqr-pow 54.5%

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

      3. pow-prod-down 60.7%

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

      4. pow2 60.7%

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

      5. *-commutative 60.7%

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

      6. metadata-eval 60.7%

         \[\leadsto d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}} \]

   if -1.999999999999994e-310 < l < 1.84999999999999998e-216

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

      1. add-sqr-sqrt 82.7%

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

      2. pow2 82.7%

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

      3. sqrt-prod 82.7%

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

      4. sqrt-pow1 82.7%

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

      5. metadata-eval 82.7%

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

      6. pow1 82.7%

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

      7. *-commutative 82.7%

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

      8. div-inv 82.7%

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

      9. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 44.9%

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

      4. neg-mul-1 44.9%

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

      5. unpow-1 44.9%

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

      6. metadata-eval 44.9%

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

      7. pow-sqr 44.9%

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

      8. rem-sqrt-square 44.9%

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

      9. rem-square-sqrt 44.9%

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

      10. fabs-sqr 44.9%

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

      11. rem-square-sqrt 44.9%

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

   9. Simplified 44.9%

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

       1. add-sqr-sqrt 44.9%

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

       2. unpow-prod-down 44.9%

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

   11. Applied egg-rr 44.9%

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

       1. pow-sqr 44.9%

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

       2. metadata-eval 44.9%

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

       3. unpow-1 44.9%

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

   13. Simplified 44.9%

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

   if 1.84999999999999998e-216 < l

   1. Initial program 62.7%

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

   3. Simplified 61.9%

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

      1. add-sqr-sqrt 61.9%

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

      2. pow2 61.9%

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

      3. sqrt-prod 61.9%

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

      4. sqrt-pow1 66.7%

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

      5. metadata-eval 66.7%

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

      6. pow1 66.7%

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

      7. *-commutative 66.7%

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

      8. div-inv 66.7%

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

      9. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in d around inf 41.9%

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

      1. unpow-1 41.9%

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

      2. metadata-eval 41.9%

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

      3. pow-sqr 41.9%

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

      4. rem-sqrt-square 41.9%

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

      5. rem-square-sqrt 41.7%

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

      6. fabs-sqr 41.7%

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

      7. rem-square-sqrt 41.9%

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

   9. Simplified 41.9%

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

       1. *-commutative 41.9%

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

       2. unpow-prod-down 49.1%

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

   11. Applied egg-rr 49.1%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.6 \cdot 10^{-246}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{2}\right)}^{-0.25}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{-254}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{elif}\;\ell \leq 1.16 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -9e-254)
     (* d (- t_0))
     (if (<= l 3.6e-308)
       (* d t_0)
       (if (<= l 1.16e-216)
         (* d (/ -1.0 (sqrt (* h l))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -9e-254) {
		tmp = d * -t_0;
	} else if (l <= 3.6e-308) {
		tmp = d * t_0;
	} else if (l <= 1.16e-216) {
		tmp = d * (-1.0 / sqrt((h * l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-9d-254)) then
        tmp = d * -t_0
    else if (l <= 3.6d-308) then
        tmp = d * t_0
    else if (l <= 1.16d-216) then
        tmp = d * ((-1.0d0) / sqrt((h * l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -9e-254) {
		tmp = d * -t_0;
	} else if (l <= 3.6e-308) {
		tmp = d * t_0;
	} else if (l <= 1.16e-216) {
		tmp = d * (-1.0 / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -9e-254:
		tmp = d * -t_0
	elif l <= 3.6e-308:
		tmp = d * t_0
	elif l <= 1.16e-216:
		tmp = d * (-1.0 / math.sqrt((h * l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -9e-254)
		tmp = Float64(d * Float64(-t_0));
	elseif (l <= 3.6e-308)
		tmp = Float64(d * t_0);
	elseif (l <= 1.16e-216)
		tmp = Float64(d * Float64(-1.0 / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -9e-254)
		tmp = d * -t_0;
	elseif (l <= 3.6e-308)
		tmp = d * t_0;
	elseif (l <= 1.16e-216)
		tmp = d * (-1.0 / sqrt((h * l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -9e-254], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[l, 3.6e-308], N[(d * t$95$0), $MachinePrecision], If[LessEqual[l, 1.16e-216], N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9e-254

   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 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. add-sqr-sqrt 65.6%

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

      2. pow2 65.6%

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

      3. sqrt-prod 65.5%

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

      4. sqrt-pow1 67.4%

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

      5. metadata-eval 67.4%

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

      6. pow1 67.4%

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

      7. *-commutative 67.4%

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

      8. div-inv 67.4%

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

      9. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 39.8%

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

      4. neg-mul-1 39.8%

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

      5. unpow-1 39.8%

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

      6. metadata-eval 39.8%

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

      7. pow-sqr 39.8%

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

      8. rem-sqrt-square 40.2%

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

      9. rem-square-sqrt 40.0%

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

      10. fabs-sqr 40.0%

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

      11. rem-square-sqrt 40.2%

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

   9. Simplified 40.2%

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

   if -9e-254 < l < 3.5999999999999999e-308

   1. Initial program 86.6%

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

   3. Simplified 86.6%

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

      1. add-sqr-sqrt 86.6%

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

      2. pow2 86.6%

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

      3. sqrt-prod 86.6%

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

      4. sqrt-pow1 86.6%

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

      5. metadata-eval 86.6%

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

      6. pow1 86.6%

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

      7. *-commutative 86.6%

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

      8. div-inv 86.6%

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

      9. metadata-eval 86.6%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in d around inf 54.5%

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

      1. unpow-1 54.5%

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

      2. metadata-eval 54.5%

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

      3. pow-sqr 54.5%

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

      4. rem-sqrt-square 54.5%

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

      5. rem-square-sqrt 54.5%

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

      6. fabs-sqr 54.5%

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

      7. rem-square-sqrt 54.5%

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

   9. Simplified 54.5%

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

   if 3.5999999999999999e-308 < l < 1.15999999999999993e-216

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

      1. add-sqr-sqrt 82.7%

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

      2. pow2 82.7%

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

      3. sqrt-prod 82.7%

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

      4. sqrt-pow1 82.7%

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

      5. metadata-eval 82.7%

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

      6. pow1 82.7%

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

      7. *-commutative 82.7%

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

      8. div-inv 82.7%

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

      9. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 44.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 44.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 44.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 44.9%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 44.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 44.9%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 44.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 44.9%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 44.9%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right)}}^{-0.5}\right) \]

       2. unpow-prod-down 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-0.5} \cdot {\left(\sqrt{h \cdot \ell}\right)}^{-0.5}}\right) \]

   11. Applied egg-rr 44.9%

       \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-0.5} \cdot {\left(\sqrt{h \cdot \ell}\right)}^{-0.5}}\right) \]
   12. Step-by-step derivation

       1. pow-sqr 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{\left(2 \cdot -0.5\right)}}\right) \]

       2. metadata-eval 44.9%

          \[\leadsto d \cdot \left(-{\left(\sqrt{h \cdot \ell}\right)}^{\color{blue}{-1}}\right) \]

       3. unpow-1 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}}\right) \]

   13. Simplified 44.9%

       \[\leadsto d \cdot \left(-\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}}\right) \]

   if 1.15999999999999993e-216 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 41.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 41.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 41.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 41.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 41.7%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 41.7%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 41.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 41.9%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 41.9%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 49.1%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 49.1%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-254}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.16 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-246} \lor \neg \left(\ell \leq -2 \cdot 10^{-310}\right) \land \ell \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (or (<= l -4.2e-246) (and (not (<= l -2e-310)) (<= l 1.25e-216)))
     (* d (- t_0))
     (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if ((l <= -4.2e-246) || (!(l <= -2e-310) && (l <= 1.25e-216))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if ((l <= (-4.2d-246)) .or. (.not. (l <= (-2d-310))) .and. (l <= 1.25d-216)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if ((l <= -4.2e-246) || (!(l <= -2e-310) && (l <= 1.25e-216))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if (l <= -4.2e-246) or (not (l <= -2e-310) and (l <= 1.25e-216)):
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if ((l <= -4.2e-246) || (!(l <= -2e-310) && (l <= 1.25e-216)))
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if ((l <= -4.2e-246) || (~((l <= -2e-310)) && (l <= 1.25e-216)))
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[Or[LessEqual[l, -4.2e-246], And[N[Not[LessEqual[l, -2e-310]], $MachinePrecision], LessEqual[l, 1.25e-216]]], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.19999999999999989e-246 or -1.999999999999994e-310 < l < 1.25000000000000005e-216

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.7%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.7%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.7%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 41.0%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.9%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 41.0%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 41.0%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -4.19999999999999989e-246 < l < -1.999999999999994e-310 or 1.25000000000000005e-216 < l

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 69.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 43.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 43.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 43.3%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 43.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 43.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 43.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 43.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 43.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 43.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-246} \lor \neg \left(\ell \leq -2 \cdot 10^{-310}\right) \land \ell \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-252}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-308}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-215}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -2e-252)
     (* d (- t_0))
     (if (<= l -3.1e-308)
       (* d t_0)
       (if (<= l 3.6e-215)
         (* d (/ -1.0 (sqrt (* h l))))
         (* d (sqrt (/ (/ 1.0 l) h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -2e-252) {
		tmp = d * -t_0;
	} else if (l <= -3.1e-308) {
		tmp = d * t_0;
	} else if (l <= 3.6e-215) {
		tmp = d * (-1.0 / sqrt((h * l)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-2d-252)) then
        tmp = d * -t_0
    else if (l <= (-3.1d-308)) then
        tmp = d * t_0
    else if (l <= 3.6d-215) then
        tmp = d * ((-1.0d0) / sqrt((h * l)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -2e-252) {
		tmp = d * -t_0;
	} else if (l <= -3.1e-308) {
		tmp = d * t_0;
	} else if (l <= 3.6e-215) {
		tmp = d * (-1.0 / Math.sqrt((h * l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -2e-252:
		tmp = d * -t_0
	elif l <= -3.1e-308:
		tmp = d * t_0
	elif l <= 3.6e-215:
		tmp = d * (-1.0 / math.sqrt((h * l)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -2e-252)
		tmp = Float64(d * Float64(-t_0));
	elseif (l <= -3.1e-308)
		tmp = Float64(d * t_0);
	elseif (l <= 3.6e-215)
		tmp = Float64(d * Float64(-1.0 / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -2e-252)
		tmp = d * -t_0;
	elseif (l <= -3.1e-308)
		tmp = d * t_0;
	elseif (l <= 3.6e-215)
		tmp = d * (-1.0 / sqrt((h * l)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -2e-252], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[l, -3.1e-308], N[(d * t$95$0), $MachinePrecision], If[LessEqual[l, 3.6e-215], N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.99999999999999989e-252

   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 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. add-sqr-sqrt 65.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 65.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 65.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 39.8%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 39.8%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 39.8%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 39.8%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 39.8%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.0%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.0%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 40.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.99999999999999989e-252 < l < -3.09999999999999992e-308

   1. Initial program 86.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 86.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 54.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 54.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 54.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 54.5%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 54.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if -3.09999999999999992e-308 < l < 3.5999999999999999e-215

   1. Initial program 82.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 82.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 44.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 44.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 44.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 44.9%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 44.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 44.9%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 44.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 44.9%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 44.9%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right)}}^{-0.5}\right) \]

       2. unpow-prod-down 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-0.5} \cdot {\left(\sqrt{h \cdot \ell}\right)}^{-0.5}}\right) \]

   11. Applied egg-rr 44.9%

       \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-0.5} \cdot {\left(\sqrt{h \cdot \ell}\right)}^{-0.5}}\right) \]
   12. Step-by-step derivation

       1. pow-sqr 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{\left(2 \cdot -0.5\right)}}\right) \]

       2. metadata-eval 44.9%

          \[\leadsto d \cdot \left(-{\left(\sqrt{h \cdot \ell}\right)}^{\color{blue}{-1}}\right) \]

       3. unpow-1 44.9%

          \[\leadsto d \cdot \left(-\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}}\right) \]

   13. Simplified 44.9%

       \[\leadsto d \cdot \left(-\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}}\right) \]

   if 3.5999999999999999e-215 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{h}{\ell}}} \cdot \sqrt{\sqrt{\frac{h}{\ell}}}\right)}\right)}^{2}\right) \]

      2. pow2 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{h}{\ell}}}\right)}^{2}}\right)}^{2}\right) \]

   8. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{h}{\ell}}}\right)}^{2}}\right)}^{2}\right) \]

   9. Taylor expanded in d around inf 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/l/ 42.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   11. Simplified 42.1%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-252}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-308}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-215}:\\ \;\;\;\;d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := d \cdot \left(-t\_0\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-309}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)) (t_1 (* d (- t_0))))
   (if (<= l -1.4e-254)
     t_1
     (if (<= l -6e-309)
       (* d t_0)
       (if (<= l 2.25e-215) t_1 (* d (sqrt (/ (/ 1.0 l) h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -1.4e-254) {
		tmp = t_1;
	} else if (l <= -6e-309) {
		tmp = d * t_0;
	} else if (l <= 2.25e-215) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    t_1 = d * -t_0
    if (l <= (-1.4d-254)) then
        tmp = t_1
    else if (l <= (-6d-309)) then
        tmp = d * t_0
    else if (l <= 2.25d-215) then
        tmp = t_1
    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 t_0 = Math.pow((h * l), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -1.4e-254) {
		tmp = t_1;
	} else if (l <= -6e-309) {
		tmp = d * t_0;
	} else if (l <= 2.25e-215) {
		tmp = t_1;
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	t_1 = d * -t_0
	tmp = 0
	if l <= -1.4e-254:
		tmp = t_1
	elif l <= -6e-309:
		tmp = d * t_0
	elif l <= 2.25e-215:
		tmp = t_1
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	t_1 = Float64(d * Float64(-t_0))
	tmp = 0.0
	if (l <= -1.4e-254)
		tmp = t_1;
	elseif (l <= -6e-309)
		tmp = Float64(d * t_0);
	elseif (l <= 2.25e-215)
		tmp = t_1;
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (h * l) ^ -0.5;
	t_1 = d * -t_0;
	tmp = 0.0;
	if (l <= -1.4e-254)
		tmp = t_1;
	elseif (l <= -6e-309)
		tmp = d * t_0;
	elseif (l <= 2.25e-215)
		tmp = t_1;
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-t$95$0)), $MachinePrecision]}, If[LessEqual[l, -1.4e-254], t$95$1, If[LessEqual[l, -6e-309], N[(d * t$95$0), $MachinePrecision], If[LessEqual[l, 2.25e-215], t$95$1, N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.39999999999999992e-254 or -6.000000000000001e-309 < l < 2.25e-215

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.7%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.7%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.7%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 41.0%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.9%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 41.0%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 41.0%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.39999999999999992e-254 < l < -6.000000000000001e-309

   1. Initial program 86.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 86.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 54.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 54.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 54.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 54.5%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 54.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 2.25e-215 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{h}{\ell}}} \cdot \sqrt{\sqrt{\frac{h}{\ell}}}\right)}\right)}^{2}\right) \]

      2. pow2 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{h}{\ell}}}\right)}^{2}}\right)}^{2}\right) \]

   8. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{h}{\ell}}}\right)}^{2}}\right)}^{2}\right) \]

   9. Taylor expanded in d around inf 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/l/ 42.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   11. Simplified 42.1%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := d \cdot \left(-t\_0\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{-250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-308}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-216}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)) (t_1 (* d (- t_0))))
   (if (<= l -7e-250)
     t_1
     (if (<= l -1.9e-308)
       (* d t_0)
       (if (<= l 8.5e-216) t_1 (* d (sqrt (/ (/ 1.0 h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -7e-250) {
		tmp = t_1;
	} else if (l <= -1.9e-308) {
		tmp = d * t_0;
	} else if (l <= 8.5e-216) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    t_1 = d * -t_0
    if (l <= (-7d-250)) then
        tmp = t_1
    else if (l <= (-1.9d-308)) then
        tmp = d * t_0
    else if (l <= 8.5d-216) then
        tmp = t_1
    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 t_0 = Math.pow((h * l), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -7e-250) {
		tmp = t_1;
	} else if (l <= -1.9e-308) {
		tmp = d * t_0;
	} else if (l <= 8.5e-216) {
		tmp = t_1;
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	t_1 = d * -t_0
	tmp = 0
	if l <= -7e-250:
		tmp = t_1
	elif l <= -1.9e-308:
		tmp = d * t_0
	elif l <= 8.5e-216:
		tmp = t_1
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	t_1 = Float64(d * Float64(-t_0))
	tmp = 0.0
	if (l <= -7e-250)
		tmp = t_1;
	elseif (l <= -1.9e-308)
		tmp = Float64(d * t_0);
	elseif (l <= 8.5e-216)
		tmp = t_1;
	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)
	t_0 = (h * l) ^ -0.5;
	t_1 = d * -t_0;
	tmp = 0.0;
	if (l <= -7e-250)
		tmp = t_1;
	elseif (l <= -1.9e-308)
		tmp = d * t_0;
	elseif (l <= 8.5e-216)
		tmp = t_1;
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-t$95$0)), $MachinePrecision]}, If[LessEqual[l, -7e-250], t$95$1, If[LessEqual[l, -1.9e-308], N[(d * t$95$0), $MachinePrecision], If[LessEqual[l, 8.5e-216], t$95$1, N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.9999999999999998e-250 or -1.9000000000000001e-308 < l < 8.50000000000000003e-216

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 68.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.7%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.7%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.7%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 41.0%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.9%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 41.0%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   9. Simplified 41.0%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -6.9999999999999998e-250 < l < -1.9000000000000001e-308

   1. Initial program 86.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. *-commutative 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. metadata-eval 86.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 86.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 54.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 54.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 54.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 54.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 54.5%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 54.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 54.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 8.50000000000000003e-216 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 42.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * ((h * l) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 66.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. metadata-eval 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. pow1 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. *-commutative 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. div-inv 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. metadata-eval 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 69.6%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 28.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 28.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 28.2%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 28.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 28.2%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 28.1%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 28.1%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 28.2%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 28.2%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))