Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 76.8%

Time: 27.4s

Alternatives: 22

Speedup: 0.5×

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: 66.7% 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: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|{\left(h \cdot \ell\right)}^{-0.5}\right| \cdot \left|d\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
       (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
      5e+285)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* (pow (/ (/ (* M D) 2.0) d) 2.0) (* 0.5 (/ h l)))))
   (* (fabs (pow (* h l) -0.5)) (fabs d))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))) <= 5e+285) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (pow((((M * D) / 2.0) / d), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = fabs(pow((h * l), -0.5)) * fabs(d);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l)))) <= 5d+285) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (((((m * d_1) / 2.0d0) / d) ** 2.0d0) * (0.5d0 * (h / l))))
    else
        tmp = abs(((h * l) ** (-0.5d0))) * abs(d)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))) <= 5e+285) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (Math.pow((((M * D) / 2.0) / d), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = Math.abs(Math.pow((h * l), -0.5)) * Math.abs(d);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))) <= 5e+285:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (math.pow((((M * D) / 2.0) / d), 2.0) * (0.5 * (h / l))))
	else:
		tmp = math.fabs(math.pow((h * l), -0.5)) * math.fabs(d)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l)))) <= 5e+285)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64((Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	else
		tmp = Float64(abs((Float64(h * l) ^ -0.5)) * abs(d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)))) <= 5e+285)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (((((M * D) / 2.0) / d) ^ 2.0) * (0.5 * (h / l))));
	else
		tmp = abs(((h * l) ^ -0.5)) * abs(d);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+285], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 5.00000000000000016e285

   1. Initial program 86.8%

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

      1. metadata-eval 86.8%

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

      2. unpow1/2 86.8%

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

      3. metadata-eval 86.8%

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

      4. unpow1/2 86.8%

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

      5. *-commutative 86.8%

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

      6. associate-*l* 86.8%

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

      7. times-frac 85.9%

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

      8. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

      1. frac-times 86.8%

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

      2. associate-/r* 86.8%

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

   5. Applied egg-rr 86.8%

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

   if 5.00000000000000016e285 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 22.9%

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

   2. Taylor expanded in d around inf 28.8%

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

      1. *-un-lft-identity 28.8%

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

      2. *-commutative 28.8%

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

   4. Applied egg-rr 28.8%

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

      1. *-lft-identity 28.8%

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

      2. associate-/r* 28.8%

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

   6. Simplified 28.8%

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

      1. add-sqr-sqrt 28.0%

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

      2. sqrt-unprod 34.1%

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

      3. pow2 34.1%

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

      4. *-commutative 34.1%

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

      5. associate-/l/ 34.1%

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

      6. inv-pow 34.1%

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

   8. Applied egg-rr 34.1%

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

      1. unpow2 34.1%

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

      2. rem-sqrt-square 57.3%

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

      3. sqr-pow 57.4%

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

      4. rem-sqrt-square 57.4%

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

      5. metadata-eval 57.4%

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

   10. Simplified 57.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|{\left(h \cdot \ell\right)}^{-0.5}\right| \cdot \left|d\right|\\ \end{array} \]

Alternative 2: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (sqrt (/ d h))
  (*
   (sqrt (/ d l))
   (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	return sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0)))))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))));
}

Python

def code(d, h, l, M, D):
	return math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))))

Julia

function code(d, h, l, M, D)
	return Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0))))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0)))));
end

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-*l* 67.8%

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

   2. metadata-eval 67.8%

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

   3. unpow1/2 67.8%

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

   4. metadata-eval 67.8%

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

   5. unpow1/2 67.8%

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

   6. associate-*l* 67.8%

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

   7. metadata-eval 67.8%

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

   8. times-frac 67.2%

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

3. Simplified 67.2%

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

4. Final simplification 67.2%

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

Alternative 3: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+156}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{\frac{\ell}{h \cdot M}} \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;\left|{\left(h \cdot \ell\right)}^{-0.5}\right| \cdot \left|d\right|\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* t_0 t_1)
          (- 1.0 (* 0.125 (/ (* (* h M) (* D (* M D))) (* l (* d d))))))))
   (if (<= d -2.1e+156)
     (*
      t_0
      (*
       t_1
       (- 1.0 (* 0.5 (* (* (/ D d) (/ D d)) (* (/ M (/ l (* h M))) 0.25))))))
     (if (<= d -6.5e+70)
       (* (fabs (pow (* h l) -0.5)) (fabs d))
       (if (<= d -4.5e-155)
         t_2
         (if (<= d -5e-310)
           (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
           (if (<= d 5.1e-163)
             (* -0.125 (* (* D (* D (* M (/ M d)))) (sqrt (/ h (pow l 3.0)))))
             (if (<= d 5.7e+88)
               t_2
               (if (<= d 8e+178)
                 (*
                  (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0))))
                  (sqrt (/ (* d (/ d h)) l)))
                 (* d (* (pow h -0.5) (pow l -0.5))))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (t_0 * t_1) * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))));
	double tmp;
	if (d <= -2.1e+156) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((D / d) * (D / d)) * ((M / (l / (h * M))) * 0.25)))));
	} else if (d <= -6.5e+70) {
		tmp = fabs(pow((h * l), -0.5)) * fabs(d);
	} else if (d <= -4.5e-155) {
		tmp = t_2;
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 5.1e-163) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * sqrt((h / pow(l, 3.0))));
	} else if (d <= 5.7e+88) {
		tmp = t_2;
	} else if (d <= 8e+178) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))) * sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (t_0 * t_1) * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))));
	double tmp;
	if (d <= -2.1e+156) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((D / d) * (D / d)) * ((M / (l / (h * M))) * 0.25)))));
	} else if (d <= -6.5e+70) {
		tmp = Math.abs(Math.pow((h * l), -0.5)) * Math.abs(d);
	} else if (d <= -4.5e-155) {
		tmp = t_2;
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 5.1e-163) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else if (d <= 5.7e+88) {
		tmp = t_2;
	} else if (d <= 8e+178) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))) * Math.sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (t_0 * t_1) * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))))
	tmp = 0
	if d <= -2.1e+156:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((D / d) * (D / d)) * ((M / (l / (h * M))) * 0.25)))))
	elif d <= -6.5e+70:
		tmp = math.fabs(math.pow((h * l), -0.5)) * math.fabs(d)
	elif d <= -4.5e-155:
		tmp = t_2
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 5.1e-163:
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * math.sqrt((h / math.pow(l, 3.0))))
	elif d <= 5.7e+88:
		tmp = t_2
	elif d <= 8e+178:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))) * math.sqrt(((d * (d / h)) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(h * M) * Float64(D * Float64(M * D))) / Float64(l * Float64(d * d))))))
	tmp = 0.0
	if (d <= -2.1e+156)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M / Float64(l / Float64(h * M))) * 0.25))))));
	elseif (d <= -6.5e+70)
		tmp = Float64(abs((Float64(h * l) ^ -0.5)) * abs(d));
	elseif (d <= -4.5e-155)
		tmp = t_2;
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 5.1e-163)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(M * Float64(M / d)))) * sqrt(Float64(h / (l ^ 3.0)))));
	elseif (d <= 5.7e+88)
		tmp = t_2;
	elseif (d <= 8e+178)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))) * sqrt(Float64(Float64(d * Float64(d / h)) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(h * M), $MachinePrecision] * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.1e+156], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M / N[(l / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.5e+70], N[(N[Abs[N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.5e-155], t$95$2, If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.1e-163], N[(-0.125 * N[(N[(D * N[(D * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.7e+88], t$95$2, If[LessEqual[d, 8e+178], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -2.09999999999999981e156

   1. Initial program 85.1%

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

      1. associate-*l* 85.1%

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

      2. metadata-eval 85.1%

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

      3. unpow1/2 85.1%

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

      4. metadata-eval 85.1%

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

      5. unpow1/2 85.1%

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

      6. associate-*l* 85.1%

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

      7. metadata-eval 85.1%

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

      8. times-frac 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 85.1%

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

      1. metadata-eval 85.1%

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

      2. *-commutative 85.1%

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

      3. times-frac 85.1%

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

      4. *-lft-identity 85.1%

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

      5. associate-/l/ 85.1%

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

      6. *-commutative 85.1%

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

      7. associate-*r/ 81.3%

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

      8. associate-/l* 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in D around 0 38.8%

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

      1. associate-/l* 38.8%

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

      2. *-commutative 38.8%

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

      3. associate-/l* 38.8%

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

      4. *-commutative 38.8%

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

      5. times-frac 35.0%

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

      6. associate-*l* 35.0%

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

      7. unpow2 35.0%

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

      8. unpow2 35.0%

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

      9. times-frac 77.6%

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

      10. unpow2 77.6%

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

      11. associate-*l* 77.6%

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

      12. associate-/l* 81.6%

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

   9. Simplified 81.6%

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

   if -2.09999999999999981e156 < d < -6.49999999999999978e70

   1. Initial program 49.8%

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

   2. Taylor expanded in d around inf 15.2%

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

      1. *-un-lft-identity 15.2%

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

      2. *-commutative 15.2%

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

   4. Applied egg-rr 15.2%

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

      1. *-lft-identity 15.2%

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

      2. associate-/r* 15.2%

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

   6. Simplified 15.2%

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

      1. add-sqr-sqrt 0.3%

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

      2. sqrt-unprod 35.8%

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

      3. pow2 35.8%

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

      4. *-commutative 35.8%

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

      5. associate-/l/ 35.8%

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

      6. inv-pow 35.8%

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

   8. Applied egg-rr 35.8%

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

      1. unpow2 35.8%

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

      2. rem-sqrt-square 71.6%

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

      3. sqr-pow 71.5%

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

      4. rem-sqrt-square 71.5%

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

      5. metadata-eval 71.5%

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

   10. Simplified 71.5%

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

   if -6.49999999999999978e70 < d < -4.5000000000000004e-155 or 5.0999999999999999e-163 < d < 5.70000000000000021e88

   1. Initial program 79.4%

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

      1. metadata-eval 79.4%

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

      2. unpow1/2 79.4%

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

      3. metadata-eval 79.4%

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

      4. unpow1/2 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l* 79.4%

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

      7. times-frac 77.1%

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

      8. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

      1. frac-times 79.4%

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

      2. associate-/r* 79.4%

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

   5. Applied egg-rr 79.4%

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

   6. Taylor expanded in M around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. *-commutative 63.4%

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

      3. unpow2 63.4%

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

      4. associate-*r* 67.0%

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

      5. unpow2 67.0%

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

      6. associate-*r* 71.7%

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

      7. *-commutative 71.7%

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

      8. unpow2 71.7%

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

      9. associate-*r* 79.6%

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

      10. *-commutative 79.6%

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

   8. Simplified 79.6%

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

   if -4.5000000000000004e-155 < d < -4.999999999999985e-310

   1. Initial program 32.2%

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

   2. Taylor expanded in d around inf 14.4%

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

      1. *-un-lft-identity 14.4%

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

      2. *-commutative 14.4%

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

   4. Applied egg-rr 14.4%

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

      1. *-lft-identity 14.4%

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

      2. unpow-1 14.4%

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

      3. sqr-pow 14.4%

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

      4. rem-sqrt-square 14.4%

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

      5. sqr-pow 14.4%

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

      6. fabs-sqr 14.4%

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

      7. sqr-pow 14.4%

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

      8. metadata-eval 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in h around -inf 8.2%

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

      1. log1p-expm1-u 33.3%

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

      2. expm1-udef 33.3%

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

   9. Applied egg-rr 49.3%

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

   if -4.999999999999985e-310 < d < 5.0999999999999999e-163

   1. Initial program 33.3%

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

      1. metadata-eval 33.3%

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

      2. unpow1/2 33.3%

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

      3. metadata-eval 33.3%

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

      4. unpow1/2 33.3%

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

      5. *-commutative 33.3%

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

      6. associate-*l* 33.3%

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

      7. times-frac 37.5%

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

      8. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

      1. frac-times 33.3%

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

      2. associate-/r* 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in d around 0 36.0%

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

      1. associate-*r/ 31.9%

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

      2. unpow2 31.9%

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

      3. unpow2 31.9%

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

      4. associate-*l* 40.5%

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

      5. associate-/l* 48.9%

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

      6. associate-/r/ 48.9%

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

   8. Simplified 48.9%

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

   if 5.70000000000000021e88 < d < 8.0000000000000004e178

   1. Initial program 94.8%

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

      1. metadata-eval 94.8%

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

      2. unpow1/2 94.8%

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

      3. metadata-eval 94.8%

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

      4. unpow1/2 94.8%

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

      5. *-commutative 94.8%

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

      6. associate-*l* 94.8%

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

      7. times-frac 94.8%

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

      8. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. frac-times 94.8%

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

      2. associate-/r* 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. sqrt-div 99.6%

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

      2. pow1/2 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

      5. associate-/r* 99.6%

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

      6. *-commutative 99.6%

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

      7. metadata-eval 99.6%

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

      8. pow1 99.6%

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

   7. Applied egg-rr 90.3%

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

      1. unpow1 90.3%

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

      2. *-commutative 90.3%

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

      3. associate-*l/ 90.4%

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

      4. associate-*r* 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 90.4%

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

   9. Simplified 90.4%

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

   if 8.0000000000000004e178 < d

   1. Initial program 69.7%

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

   2. Taylor expanded in d around inf 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. *-commutative 71.7%

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

   4. Applied egg-rr 71.7%

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

      1. *-lft-identity 71.7%

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

      2. unpow-1 71.7%

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

      3. sqr-pow 71.8%

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

      4. rem-sqrt-square 71.8%

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

      5. sqr-pow 71.6%

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

      6. fabs-sqr 71.6%

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

      7. sqr-pow 71.8%

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

      8. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 7 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{\frac{\ell}{h \cdot M}} \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;\left|{\left(h \cdot \ell\right)}^{-0.5}\right| \cdot \left|d\right|\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 55.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{if}\;d \leq -1.46 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{elif}\;d \leq 2400000 \lor \neg \left(d \leq 7.2 \cdot 10^{+84}\right) \land d \leq 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0))))
          (sqrt (/ (* d (/ d h)) l)))))
   (if (<= d -1.46e-112)
     t_0
     (if (<= d -5e-310)
       (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
       (if (<= d 1.75e-93)
         (* (/ (sqrt h) (pow l 1.5)) (/ -0.125 (/ d (* M (* M (* D D))))))
         (if (or (<= d 2400000.0) (and (not (<= d 7.2e+84)) (<= d 1e+178)))
           t_0
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))) * sqrt(((d * (d / h)) / l));
	double tmp;
	if (d <= -1.46e-112) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.75e-93) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else if ((d <= 2400000.0) || (!(d <= 7.2e+84) && (d <= 1e+178))) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))) * Math.sqrt(((d * (d / h)) / l));
	double tmp;
	if (d <= -1.46e-112) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.75e-93) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else if ((d <= 2400000.0) || (!(d <= 7.2e+84) && (d <= 1e+178))) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))) * math.sqrt(((d * (d / h)) / l))
	tmp = 0
	if d <= -1.46e-112:
		tmp = t_0
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 1.75e-93:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))))
	elif (d <= 2400000.0) or (not (d <= 7.2e+84) and (d <= 1e+178)):
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))) * sqrt(Float64(Float64(d * Float64(d / h)) / l)))
	tmp = 0.0
	if (d <= -1.46e-112)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 1.75e-93)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 / Float64(d / Float64(M * Float64(M * Float64(D * D))))));
	elseif ((d <= 2400000.0) || (!(d <= 7.2e+84) && (d <= 1e+178)))
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.46e-112], t$95$0, If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-93], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d / N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 2400000.0], And[N[Not[LessEqual[d, 7.2e+84]], $MachinePrecision], LessEqual[d, 1e+178]]], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.46000000000000002e-112 or 1.75e-93 < d < 2.4e6 or 7.1999999999999999e84 < d < 1.0000000000000001e178

   1. Initial program 77.5%

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

      1. metadata-eval 77.5%

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

      2. unpow1/2 77.5%

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

      3. metadata-eval 77.5%

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

      4. unpow1/2 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*l* 77.5%

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

      7. times-frac 76.7%

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

      8. metadata-eval 76.7%

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

   3. Simplified 76.7%

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

      1. frac-times 77.5%

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

      2. associate-/r* 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. sqrt-div 34.5%

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

      2. pow1/2 34.5%

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

      3. metadata-eval 34.5%

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

      4. associate-*r* 34.5%

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

      5. associate-/r* 34.5%

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

      6. *-commutative 34.5%

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

      7. metadata-eval 34.5%

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

      8. pow1 34.5%

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

   7. Applied egg-rr 71.0%

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

      1. unpow1 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*l/ 68.8%

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

      4. associate-*r* 68.8%

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

      5. *-commutative 68.8%

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

      6. associate-*l* 68.8%

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

   9. Simplified 68.8%

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

   if -1.46000000000000002e-112 < d < -4.999999999999985e-310

   1. Initial program 43.3%

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

   2. Taylor expanded in d around inf 11.3%

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

      1. *-un-lft-identity 11.3%

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

      2. *-commutative 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. *-lft-identity 11.3%

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

      2. unpow-1 11.3%

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

      3. sqr-pow 11.3%

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

      4. rem-sqrt-square 11.3%

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

      5. sqr-pow 11.3%

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

      6. fabs-sqr 11.3%

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

      7. sqr-pow 11.3%

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

      8. metadata-eval 11.3%

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

   6. Simplified 11.3%

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

   7. Taylor expanded in h around -inf 6.8%

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

      1. log1p-expm1-u 27.2%

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

      2. expm1-udef 27.2%

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

   9. Applied egg-rr 47.9%

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

   if -4.999999999999985e-310 < d < 1.75e-93

   1. Initial program 54.5%

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

   2. Taylor expanded in d around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r/ 43.0%

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

      4. associate-/l* 43.0%

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

      5. unpow2 43.0%

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

      6. associate-*r* 48.4%

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

      7. unpow2 48.4%

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

   4. Simplified 48.4%

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

      1. sqrt-div 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. sqr-pow 53.8%

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

      2. rem-sqrt-square 56.4%

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

      3. sqr-pow 56.3%

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

      4. fabs-sqr 56.3%

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

      5. sqr-pow 56.4%

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

      6. metadata-eval 56.4%

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

   8. Simplified 56.4%

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

   if 2.4e6 < d < 7.1999999999999999e84 or 1.0000000000000001e178 < d

   1. Initial program 74.7%

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

   2. Taylor expanded in d around inf 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. *-commutative 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. *-lft-identity 63.2%

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

      2. unpow-1 63.2%

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

      3. sqr-pow 63.2%

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

      4. rem-sqrt-square 63.2%

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

      5. sqr-pow 63.0%

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

      6. fabs-sqr 63.0%

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

      7. sqr-pow 63.2%

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

      8. metadata-eval 63.2%

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

   6. Simplified 63.2%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.46 \cdot 10^{-112}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{elif}\;d \leq 2400000 \lor \neg \left(d \leq 7.2 \cdot 10^{+84}\right) \land d \leq 10^{+178}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-153}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+86}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= d -2.9e-153)
     (* t_0 (- 1.0 (* 0.125 (* (* M (/ M d)) (* (/ h d) (* D (/ D l)))))))
     (if (<= d -5e-310)
       (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
       (if (<= d 1.55e-91)
         (* (/ (sqrt h) (pow l 1.5)) (/ -0.125 (/ d (* M (* M (* D D))))))
         (if (<= d 1.08e+86)
           (*
            t_0
            (- 1.0 (* 0.125 (* (/ D l) (/ (* D (* M M)) (/ (* d d) h))))))
           (if (<= d 7.2e+189)
             (*
              (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0))))
              (sqrt (/ (* d (/ d h)) l)))
             (* d (* (pow h -0.5) (pow l -0.5))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (d <= -2.9e-153) {
		tmp = t_0 * (1.0 - (0.125 * ((M * (M / d)) * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.55e-91) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else if (d <= 1.08e+86) {
		tmp = t_0 * (1.0 - (0.125 * ((D / l) * ((D * (M * M)) / ((d * d) / h)))));
	} else if (d <= 7.2e+189) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))) * sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (d <= -2.9e-153) {
		tmp = t_0 * (1.0 - (0.125 * ((M * (M / d)) * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.55e-91) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else if (d <= 1.08e+86) {
		tmp = t_0 * (1.0 - (0.125 * ((D / l) * ((D * (M * M)) / ((d * d) / h)))));
	} else if (d <= 7.2e+189) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))) * Math.sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if d <= -2.9e-153:
		tmp = t_0 * (1.0 - (0.125 * ((M * (M / d)) * ((h / d) * (D * (D / l))))))
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 1.55e-91:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))))
	elif d <= 1.08e+86:
		tmp = t_0 * (1.0 - (0.125 * ((D / l) * ((D * (M * M)) / ((d * d) / h)))))
	elif d <= 7.2e+189:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))) * math.sqrt(((d * (d / h)) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (d <= -2.9e-153)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(M * Float64(M / d)) * Float64(Float64(h / d) * Float64(D * Float64(D / l)))))));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 1.55e-91)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 / Float64(d / Float64(M * Float64(M * Float64(D * D))))));
	elseif (d <= 1.08e+86)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(D / l) * Float64(Float64(D * Float64(M * M)) / Float64(Float64(d * d) / h))))));
	elseif (d <= 7.2e+189)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))) * sqrt(Float64(Float64(d * Float64(d / h)) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-153], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-91], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d / N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.08e+86], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(D / l), $MachinePrecision] * N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+189], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -2.90000000000000002e-153

   1. Initial program 72.4%

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

      1. metadata-eval 72.4%

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

      2. unpow1/2 72.4%

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

      3. metadata-eval 72.4%

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

      4. unpow1/2 72.4%

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

      5. *-commutative 72.4%

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

      6. associate-*l* 72.4%

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

      7. times-frac 70.8%

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

      8. metadata-eval 70.8%

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

   3. Simplified 70.8%

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

      1. frac-times 72.4%

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

      2. associate-/r* 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in M around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 50.3%

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

      3. *-commutative 50.3%

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

      4. unpow2 50.3%

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

      5. associate-*r* 51.5%

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

      6. unpow2 51.5%

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

      7. associate-*l/ 54.1%

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

      8. associate-*r* 53.0%

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

      9. unpow2 53.0%

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

      10. unpow2 53.0%

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

      11. times-frac 61.1%

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

      12. unpow2 61.1%

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

      13. associate-/r/ 61.1%

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

      14. *-commutative 61.1%

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

      15. associate-*l* 63.2%

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

      16. associate-/l* 65.4%

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

   8. Simplified 65.4%

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

   if -2.90000000000000002e-153 < d < -4.999999999999985e-310

   1. Initial program 32.2%

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

   2. Taylor expanded in d around inf 14.4%

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

      1. *-un-lft-identity 14.4%

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

      2. *-commutative 14.4%

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

   4. Applied egg-rr 14.4%

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

      1. *-lft-identity 14.4%

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

      2. unpow-1 14.4%

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

      3. sqr-pow 14.4%

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

      4. rem-sqrt-square 14.4%

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

      5. sqr-pow 14.4%

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

      6. fabs-sqr 14.4%

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

      7. sqr-pow 14.4%

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

      8. metadata-eval 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in h around -inf 8.2%

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

      1. log1p-expm1-u 33.3%

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

      2. expm1-udef 33.3%

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

   9. Applied egg-rr 49.3%

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

   if -4.999999999999985e-310 < d < 1.5499999999999999e-91

   1. Initial program 55.6%

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

   2. Taylor expanded in d around 0 41.9%

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

      1. associate-*r* 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-*r/ 41.9%

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

      4. associate-/l* 41.9%

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

      5. unpow2 41.9%

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

      6. associate-*r* 47.3%

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

      7. unpow2 47.3%

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

   4. Simplified 47.3%

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

      1. sqrt-div 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. sqr-pow 52.4%

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

      2. rem-sqrt-square 55.0%

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

      3. sqr-pow 54.9%

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

      4. fabs-sqr 54.9%

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

      5. sqr-pow 55.0%

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

      6. metadata-eval 55.0%

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

   8. Simplified 55.0%

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

   if 1.5499999999999999e-91 < d < 1.07999999999999993e86

   1. Initial program 80.3%

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

      1. metadata-eval 80.3%

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

      2. unpow1/2 80.3%

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

      3. metadata-eval 80.3%

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

      4. unpow1/2 80.3%

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

      5. *-commutative 80.3%

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

      6. associate-*l* 80.3%

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

      7. times-frac 78.2%

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

      8. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

      1. frac-times 80.3%

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

      2. associate-/r* 80.3%

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

   5. Applied egg-rr 80.3%

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

   6. Taylor expanded in M around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. times-frac 62.9%

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

      3. *-commutative 62.9%

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

      4. unpow2 62.9%

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

      5. associate-*r* 65.0%

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

      6. unpow2 65.0%

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

      7. associate-*l/ 69.6%

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

      8. associate-*r* 67.4%

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

      9. unpow2 67.4%

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

      10. unpow2 67.4%

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

      11. times-frac 65.1%

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

      12. unpow2 65.1%

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

      13. associate-/r/ 65.1%

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

      14. *-commutative 65.1%

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

      15. associate-*l* 64.9%

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

      16. associate-/l* 65.0%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in M around 0 65.1%

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

       1. unpow2 65.1%

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

       2. unpow2 65.1%

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

       3. times-frac 62.9%

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

       4. associate-*l/ 67.4%

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

       5. unpow2 67.4%

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

       6. *-commutative 67.4%

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

       7. associate-*l* 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-/l* 74.0%

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

       10. associate-*l/ 74.0%

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

   11. Simplified 74.0%

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

   if 1.07999999999999993e86 < d < 7.20000000000000017e189

   1. Initial program 94.8%

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

      1. metadata-eval 94.8%

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

      2. unpow1/2 94.8%

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

      3. metadata-eval 94.8%

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

      4. unpow1/2 94.8%

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

      5. *-commutative 94.8%

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

      6. associate-*l* 94.8%

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

      7. times-frac 94.8%

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

      8. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. frac-times 94.8%

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

      2. associate-/r* 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. sqrt-div 99.6%

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

      2. pow1/2 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

      5. associate-/r* 99.6%

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

      6. *-commutative 99.6%

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

      7. metadata-eval 99.6%

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

      8. pow1 99.6%

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

   7. Applied egg-rr 90.3%

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

      1. unpow1 90.3%

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

      2. *-commutative 90.3%

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

      3. associate-*l/ 90.4%

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

      4. associate-*r* 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 90.4%

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

   9. Simplified 90.4%

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

   if 7.20000000000000017e189 < d

   1. Initial program 69.7%

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

   2. Taylor expanded in d around inf 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. *-commutative 71.7%

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

   4. Applied egg-rr 71.7%

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

      1. *-lft-identity 71.7%

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

      2. unpow-1 71.7%

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

      3. sqr-pow 71.8%

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

      4. rem-sqrt-square 71.8%

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

      5. sqr-pow 71.6%

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

      6. fabs-sqr 71.6%

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

      7. sqr-pow 71.8%

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

      8. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-153}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+86}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \left(t_0 \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot t_0\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ M d))) (t_1 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= d -1.45e-155)
     (* t_1 (- 1.0 (* 0.125 (* t_0 (* (/ h d) (* D (/ D l)))))))
     (if (<= d -5e-310)
       (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
       (if (<= d 5.1e-163)
         (* -0.125 (* (* D (* D t_0)) (sqrt (/ h (pow l 3.0)))))
         (if (<= d 3.5e+85)
           (*
            t_1
            (- 1.0 (* 0.125 (/ (* (* h M) (* D (* M D))) (* l (* d d))))))
           (if (<= d 2e+183)
             (*
              (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0))))
              (sqrt (/ (* d (/ d h)) l)))
             (* d (* (pow h -0.5) (pow l -0.5))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (M / d);
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (d <= -1.45e-155) {
		tmp = t_1 * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 5.1e-163) {
		tmp = -0.125 * ((D * (D * t_0)) * sqrt((h / pow(l, 3.0))));
	} else if (d <= 3.5e+85) {
		tmp = t_1 * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))));
	} else if (d <= 2e+183) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))) * sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (M / d);
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (d <= -1.45e-155) {
		tmp = t_1 * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 5.1e-163) {
		tmp = -0.125 * ((D * (D * t_0)) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else if (d <= 3.5e+85) {
		tmp = t_1 * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))));
	} else if (d <= 2e+183) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))) * Math.sqrt(((d * (d / h)) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (M / d)
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if d <= -1.45e-155:
		tmp = t_1 * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))))
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 5.1e-163:
		tmp = -0.125 * ((D * (D * t_0)) * math.sqrt((h / math.pow(l, 3.0))))
	elif d <= 3.5e+85:
		tmp = t_1 * (1.0 - (0.125 * (((h * M) * (D * (M * D))) / (l * (d * d)))))
	elif d <= 2e+183:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))) * math.sqrt(((d * (d / h)) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(M / d))
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (d <= -1.45e-155)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.125 * Float64(t_0 * Float64(Float64(h / d) * Float64(D * Float64(D / l)))))));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 5.1e-163)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D * t_0)) * sqrt(Float64(h / (l ^ 3.0)))));
	elseif (d <= 3.5e+85)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(h * M) * Float64(D * Float64(M * D))) / Float64(l * Float64(d * d))))));
	elseif (d <= 2e+183)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))) * sqrt(Float64(Float64(d * Float64(d / h)) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.45e-155], N[(t$95$1 * N[(1.0 - N[(0.125 * N[(t$95$0 * N[(N[(h / d), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.1e-163], N[(-0.125 * N[(N[(D * N[(D * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+85], N[(t$95$1 * N[(1.0 - N[(0.125 * N[(N[(N[(h * M), $MachinePrecision] * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+183], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.45000000000000005e-155

   1. Initial program 72.4%

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

      1. metadata-eval 72.4%

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

      2. unpow1/2 72.4%

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

      3. metadata-eval 72.4%

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

      4. unpow1/2 72.4%

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

      5. *-commutative 72.4%

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

      6. associate-*l* 72.4%

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

      7. times-frac 70.8%

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

      8. metadata-eval 70.8%

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

   3. Simplified 70.8%

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

      1. frac-times 72.4%

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

      2. associate-/r* 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in M around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 50.3%

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

      3. *-commutative 50.3%

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

      4. unpow2 50.3%

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

      5. associate-*r* 51.5%

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

      6. unpow2 51.5%

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

      7. associate-*l/ 54.1%

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

      8. associate-*r* 53.0%

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

      9. unpow2 53.0%

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

      10. unpow2 53.0%

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

      11. times-frac 61.1%

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

      12. unpow2 61.1%

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

      13. associate-/r/ 61.1%

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

      14. *-commutative 61.1%

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

      15. associate-*l* 63.2%

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

      16. associate-/l* 65.4%

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

   8. Simplified 65.4%

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

   if -1.45000000000000005e-155 < d < -4.999999999999985e-310

   1. Initial program 32.2%

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

   2. Taylor expanded in d around inf 14.4%

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

      1. *-un-lft-identity 14.4%

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

      2. *-commutative 14.4%

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

   4. Applied egg-rr 14.4%

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

      1. *-lft-identity 14.4%

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

      2. unpow-1 14.4%

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

      3. sqr-pow 14.4%

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

      4. rem-sqrt-square 14.4%

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

      5. sqr-pow 14.4%

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

      6. fabs-sqr 14.4%

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

      7. sqr-pow 14.4%

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

      8. metadata-eval 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in h around -inf 8.2%

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

      1. log1p-expm1-u 33.3%

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

      2. expm1-udef 33.3%

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

   9. Applied egg-rr 49.3%

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

   if -4.999999999999985e-310 < d < 5.0999999999999999e-163

   1. Initial program 33.3%

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

      1. metadata-eval 33.3%

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

      2. unpow1/2 33.3%

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

      3. metadata-eval 33.3%

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

      4. unpow1/2 33.3%

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

      5. *-commutative 33.3%

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

      6. associate-*l* 33.3%

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

      7. times-frac 37.5%

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

      8. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

      1. frac-times 33.3%

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

      2. associate-/r* 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in d around 0 36.0%

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

      1. associate-*r/ 31.9%

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

      2. unpow2 31.9%

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

      3. unpow2 31.9%

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

      4. associate-*l* 40.5%

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

      5. associate-/l* 48.9%

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

      6. associate-/r/ 48.9%

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

   8. Simplified 48.9%

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

   if 5.0999999999999999e-163 < d < 3.50000000000000005e85

   1. Initial program 82.3%

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

      1. metadata-eval 82.3%

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

      2. unpow1/2 82.3%

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

      3. metadata-eval 82.3%

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

      4. unpow1/2 82.3%

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

      5. *-commutative 82.3%

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

      6. associate-*l* 82.3%

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

      7. times-frac 80.7%

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

      8. metadata-eval 80.7%

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

   3. Simplified 80.7%

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

      1. frac-times 82.3%

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

      2. associate-/r* 82.3%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in M around 0 61.2%

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

      1. *-commutative 61.2%

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

      2. *-commutative 61.2%

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

      3. unpow2 61.2%

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

      4. associate-*r* 64.5%

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

      5. unpow2 64.5%

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

      6. associate-*r* 69.4%

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

      7. *-commutative 69.4%

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

      8. unpow2 69.4%

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

      9. associate-*r* 79.2%

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

      10. *-commutative 79.2%

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

   8. Simplified 79.2%

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

   if 3.50000000000000005e85 < d < 1.99999999999999989e183

   1. Initial program 94.8%

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

      1. metadata-eval 94.8%

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

      2. unpow1/2 94.8%

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

      3. metadata-eval 94.8%

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

      4. unpow1/2 94.8%

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

      5. *-commutative 94.8%

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

      6. associate-*l* 94.8%

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

      7. times-frac 94.8%

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

      8. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. frac-times 94.8%

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

      2. associate-/r* 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. sqrt-div 99.6%

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

      2. pow1/2 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

      5. associate-/r* 99.6%

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

      6. *-commutative 99.6%

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

      7. metadata-eval 99.6%

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

      8. pow1 99.6%

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

   7. Applied egg-rr 90.3%

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

      1. unpow1 90.3%

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

      2. *-commutative 90.3%

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

      3. associate-*l/ 90.4%

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

      4. associate-*r* 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 90.4%

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

   9. Simplified 90.4%

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

   if 1.99999999999999989e183 < d

   1. Initial program 69.7%

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

   2. Taylor expanded in d around inf 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. *-commutative 71.7%

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

   4. Applied egg-rr 71.7%

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

      1. *-lft-identity 71.7%

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

      2. unpow-1 71.7%

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

      3. sqr-pow 71.8%

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

      4. rem-sqrt-square 71.8%

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

      5. sqr-pow 71.6%

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

      6. fabs-sqr 71.6%

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

      7. sqr-pow 71.8%

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

      8. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (/ (/ M d) 2.0)) 2.0))))
          (sqrt (* (/ d h) (/ d l))))))
   (if (<= d -3.8e-112)
     t_0
     (if (<= d -5e-310)
       (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
       (if (<= d 1.02e-163)
         (* -0.125 (* (* D (* D (* M (/ M d)))) (sqrt (/ h (pow l 3.0)))))
         (if (<= d 1.85e+179) t_0 (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * ((h / l) * pow((D * ((M / d) / 2.0)), 2.0)))) * sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -3.8e-112) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.02e-163) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * sqrt((h / pow(l, 3.0))));
	} else if (d <= 1.85e+179) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * ((h / l) * Math.pow((D * ((M / d) / 2.0)), 2.0)))) * Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -3.8e-112) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.02e-163) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else if (d <= 1.85e+179) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 + (-0.5 * ((h / l) * math.pow((D * ((M / d) / 2.0)), 2.0)))) * math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -3.8e-112:
		tmp = t_0
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 1.02e-163:
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * math.sqrt((h / math.pow(l, 3.0))))
	elif d <= 1.85e+179:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))))
	tmp = 0.0
	if (d <= -3.8e-112)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 1.02e-163)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(M * Float64(M / d)))) * sqrt(Float64(h / (l ^ 3.0)))));
	elseif (d <= 1.85e+179)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e-112], t$95$0, If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-163], N[(-0.125 * N[(N[(D * N[(D * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+179], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.79999999999999995e-112 or 1.02000000000000007e-163 < d < 1.85e179

   1. Initial program 78.9%

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

   3. Applied egg-rr 20.6%

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

      1. expm1-def 35.3%

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

      2. expm1-log1p 69.0%

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

      3. cancel-sign-sub-inv 69.0%

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

      4. metadata-eval 69.0%

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

      5. associate-*l/ 71.0%

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

      6. *-commutative 71.0%

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

      7. associate-/l* 69.0%

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

      8. associate-*r/ 69.0%

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

      9. *-commutative 69.0%

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

      10. associate-*r/ 69.0%

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

      11. +-commutative 69.0%

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

      12. fma-def 69.0%

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

   5. Simplified 69.0%

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

   if -3.79999999999999995e-112 < d < -4.999999999999985e-310

   1. Initial program 43.3%

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

   2. Taylor expanded in d around inf 11.3%

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

      1. *-un-lft-identity 11.3%

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

      2. *-commutative 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. *-lft-identity 11.3%

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

      2. unpow-1 11.3%

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

      3. sqr-pow 11.3%

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

      4. rem-sqrt-square 11.3%

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

      5. sqr-pow 11.3%

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

      6. fabs-sqr 11.3%

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

      7. sqr-pow 11.3%

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

      8. metadata-eval 11.3%

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

   6. Simplified 11.3%

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

   7. Taylor expanded in h around -inf 6.8%

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

      1. log1p-expm1-u 27.2%

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

      2. expm1-udef 27.2%

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

   9. Applied egg-rr 47.9%

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

   if -4.999999999999985e-310 < d < 1.02000000000000007e-163

   1. Initial program 33.3%

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

      1. metadata-eval 33.3%

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

      2. unpow1/2 33.3%

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

      3. metadata-eval 33.3%

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

      4. unpow1/2 33.3%

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

      5. *-commutative 33.3%

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

      6. associate-*l* 33.3%

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

      7. times-frac 37.5%

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

      8. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

      1. frac-times 33.3%

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

      2. associate-/r* 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in d around 0 36.0%

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

      1. associate-*r/ 31.9%

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

      2. unpow2 31.9%

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

      3. unpow2 31.9%

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

      4. associate-*l* 40.5%

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

      5. associate-/l* 48.9%

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

      6. associate-/r/ 48.9%

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

   8. Simplified 48.9%

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

   if 1.85e179 < d

   1. Initial program 69.7%

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

   2. Taylor expanded in d around inf 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. *-commutative 71.7%

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

   4. Applied egg-rr 71.7%

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

      1. *-lft-identity 71.7%

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

      2. unpow-1 71.7%

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

      3. sqr-pow 71.8%

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

      4. rem-sqrt-square 71.8%

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

      5. sqr-pow 71.6%

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

      6. fabs-sqr 71.6%

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

      7. sqr-pow 71.8%

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

      8. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-112}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 59.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{M}{d}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-157}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(t_0 \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot t_0\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+178}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ M d))))
   (if (<= d -3.4e-157)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (* 0.125 (* t_0 (* (/ h d) (* D (/ D l)))))))
     (if (<= d -5e-310)
       (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
       (if (<= d 1.08e-163)
         (* -0.125 (* (* D (* D t_0)) (sqrt (/ h (pow l 3.0)))))
         (if (<= d 3.05e+178)
           (*
            (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (/ (/ M d) 2.0)) 2.0))))
            (sqrt (* (/ d h) (/ d l))))
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (M / d);
	double tmp;
	if (d <= -3.4e-157) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.08e-163) {
		tmp = -0.125 * ((D * (D * t_0)) * sqrt((h / pow(l, 3.0))));
	} else if (d <= 3.05e+178) {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((D * ((M / d) / 2.0)), 2.0)))) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (M / d);
	double tmp;
	if (d <= -3.4e-157) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.08e-163) {
		tmp = -0.125 * ((D * (D * t_0)) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else if (d <= 3.05e+178) {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((D * ((M / d) / 2.0)), 2.0)))) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (M / d)
	tmp = 0
	if d <= -3.4e-157:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.125 * (t_0 * ((h / d) * (D * (D / l))))))
	elif d <= -5e-310:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	elif d <= 1.08e-163:
		tmp = -0.125 * ((D * (D * t_0)) * math.sqrt((h / math.pow(l, 3.0))))
	elif d <= 3.05e+178:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((D * ((M / d) / 2.0)), 2.0)))) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(M / d))
	tmp = 0.0
	if (d <= -3.4e-157)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(t_0 * Float64(Float64(h / d) * Float64(D * Float64(D / l)))))));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 1.08e-163)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D * t_0)) * sqrt(Float64(h / (l ^ 3.0)))));
	elseif (d <= 3.05e+178)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.4e-157], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(t$95$0 * N[(N[(h / d), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.08e-163], N[(-0.125 * N[(N[(D * N[(D * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e+178], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.39999999999999977e-157

   1. Initial program 72.4%

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

      1. metadata-eval 72.4%

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

      2. unpow1/2 72.4%

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

      3. metadata-eval 72.4%

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

      4. unpow1/2 72.4%

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

      5. *-commutative 72.4%

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

      6. associate-*l* 72.4%

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

      7. times-frac 70.8%

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

      8. metadata-eval 70.8%

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

   3. Simplified 70.8%

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

      1. frac-times 72.4%

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

      2. associate-/r* 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in M around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 50.3%

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

      3. *-commutative 50.3%

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

      4. unpow2 50.3%

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

      5. associate-*r* 51.5%

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

      6. unpow2 51.5%

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

      7. associate-*l/ 54.1%

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

      8. associate-*r* 53.0%

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

      9. unpow2 53.0%

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

      10. unpow2 53.0%

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

      11. times-frac 61.1%

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

      12. unpow2 61.1%

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

      13. associate-/r/ 61.1%

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

      14. *-commutative 61.1%

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

      15. associate-*l* 63.2%

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

      16. associate-/l* 65.4%

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

   8. Simplified 65.4%

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

   if -3.39999999999999977e-157 < d < -4.999999999999985e-310

   1. Initial program 32.2%

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

   2. Taylor expanded in d around inf 14.4%

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

      1. *-un-lft-identity 14.4%

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

      2. *-commutative 14.4%

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

   4. Applied egg-rr 14.4%

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

      1. *-lft-identity 14.4%

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

      2. unpow-1 14.4%

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

      3. sqr-pow 14.4%

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

      4. rem-sqrt-square 14.4%

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

      5. sqr-pow 14.4%

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

      6. fabs-sqr 14.4%

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

      7. sqr-pow 14.4%

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

      8. metadata-eval 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in h around -inf 8.2%

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

      1. log1p-expm1-u 33.3%

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

      2. expm1-udef 33.3%

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

   9. Applied egg-rr 49.3%

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

   if -4.999999999999985e-310 < d < 1.0800000000000001e-163

   1. Initial program 33.3%

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

      1. metadata-eval 33.3%

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

      2. unpow1/2 33.3%

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

      3. metadata-eval 33.3%

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

      4. unpow1/2 33.3%

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

      5. *-commutative 33.3%

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

      6. associate-*l* 33.3%

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

      7. times-frac 37.5%

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

      8. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

      1. frac-times 33.3%

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

      2. associate-/r* 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in d around 0 36.0%

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

      1. associate-*r/ 31.9%

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

      2. unpow2 31.9%

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

      3. unpow2 31.9%

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

      4. associate-*l* 40.5%

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

      5. associate-/l* 48.9%

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

      6. associate-/r/ 48.9%

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

   8. Simplified 48.9%

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

   if 1.0800000000000001e-163 < d < 3.05e178

   1. Initial program 85.4%

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

   3. Applied egg-rr 19.3%

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

      1. expm1-def 37.1%

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

      2. expm1-log1p 71.4%

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

      3. cancel-sign-sub-inv 71.4%

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

      4. metadata-eval 71.4%

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

      5. associate-*l/ 72.7%

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

      6. *-commutative 72.7%

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

      7. associate-/l* 71.3%

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

      8. associate-*r/ 71.3%

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

      9. *-commutative 71.3%

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

      10. associate-*r/ 71.3%

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

      11. +-commutative 71.3%

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

      12. fma-def 71.3%

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

   5. Simplified 71.4%

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

   if 3.05e178 < d

   1. Initial program 69.7%

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

   2. Taylor expanded in d around inf 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. *-commutative 71.7%

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

   4. Applied egg-rr 71.7%

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

      1. *-lft-identity 71.7%

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

      2. unpow-1 71.7%

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

      3. sqr-pow 71.8%

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

      4. rem-sqrt-square 71.8%

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

      5. sqr-pow 71.6%

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

      6. fabs-sqr 71.6%

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

      7. sqr-pow 71.8%

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

      8. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. unpow-prod-down 81.9%

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

   8. Applied egg-rr 81.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{-157}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{-163}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+178}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;d \leq -6 \cdot 10^{-52}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* h l))))
   (if (<= d -6e-52)
     (* d (cbrt (* (/ -1.0 (* h l)) (sqrt (/ (/ 1.0 h) l)))))
     (if (<= d -1.7e-71)
       (* d (sqrt (cbrt (* t_0 (* t_0 t_0)))))
       (if (<= d -2.1e-94)
         (* (sqrt (/ d h)) (sqrt (/ d l)))
         (if (<= d -5e-310)
           (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
           (if (<= d 1.1e-92)
             (* (/ (sqrt h) (pow l 1.5)) (/ -0.125 (/ d (* M (* M (* D D))))))
             (* d (* (pow h -0.5) (pow l -0.5))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (d <= -6e-52) {
		tmp = d * cbrt(((-1.0 / (h * l)) * sqrt(((1.0 / h) / l))));
	} else if (d <= -1.7e-71) {
		tmp = d * sqrt(cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= -2.1e-94) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.1e-92) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (d <= -6e-52) {
		tmp = d * Math.cbrt(((-1.0 / (h * l)) * Math.sqrt(((1.0 / h) / l))));
	} else if (d <= -1.7e-71) {
		tmp = d * Math.sqrt(Math.cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= -2.1e-94) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 1.1e-92) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (d <= -6e-52)
		tmp = Float64(d * cbrt(Float64(Float64(-1.0 / Float64(h * l)) * sqrt(Float64(Float64(1.0 / h) / l)))));
	elseif (d <= -1.7e-71)
		tmp = Float64(d * sqrt(cbrt(Float64(t_0 * Float64(t_0 * t_0)))));
	elseif (d <= -2.1e-94)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 1.1e-92)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 / Float64(d / Float64(M * Float64(M * Float64(D * D))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6e-52], N[(d * N[Power[N[(N[(-1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.7e-71], N[(d * N[Sqrt[N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.1e-94], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-92], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d / N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -6e-52

   1. Initial program 71.7%

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

   2. Taylor expanded in d around inf 10.7%

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

      1. add-cbrt-cube 13.6%

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

      2. *-commutative 13.6%

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

      3. *-commutative 13.6%

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

      4. *-commutative 13.6%

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

   4. Applied egg-rr 13.6%

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

      1. add-cbrt-cube 13.6%

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

      2. add-sqr-sqrt 13.6%

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

      3. add-cbrt-cube 13.6%

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

      4. associate-/r* 13.6%

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

      5. add-cbrt-cube 12.2%

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

      6. associate-/r* 12.2%

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

   6. Applied egg-rr 12.2%

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

      1. div-inv 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. mul-1-neg 12.2%

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

      5. add-sqr-sqrt 12.2%

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

      6. sqrt-unprod 22.2%

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

      7. mul-1-neg 22.2%

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

      8. mul-1-neg 22.2%

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

      9. sqr-neg 22.2%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 47.8%

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

   8. Applied egg-rr 47.8%

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

      1. associate-*r/ 47.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h} \cdot -1}{\ell}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. associate-*l/ 47.8%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1 \cdot -1}{h}}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 47.8%

         \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{-1}}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. associate-/l/ 47.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   10. Simplified 47.8%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -6e-52 < d < -1.70000000000000002e-71

   1. Initial program 80.0%

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

   2. Taylor expanded in d around inf 4.3%

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

      1. add-cbrt-cube 81.6%

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

      2. *-commutative 81.6%

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

      3. *-commutative 81.6%

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

      4. *-commutative 81.6%

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

   4. Applied egg-rr 81.6%

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

   if -1.70000000000000002e-71 < d < -2.1000000000000001e-94

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow1/2 59.7%

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

      4. metadata-eval 59.7%

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

      5. unpow1/2 59.7%

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

      6. sub-neg 59.7%

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

      7. +-commutative 59.7%

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

      8. *-commutative 59.7%

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

      9. distribute-rgt-neg-in 59.7%

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

      10. fma-def 59.7%

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

   3. Simplified 40.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 60.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -2.1000000000000001e-94 < d < -4.999999999999985e-310

   1. Initial program 46.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 12.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 12.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 12.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 12.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 12.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 12.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 12.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 12.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 12.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 12.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.7%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 29.9%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 29.9%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 47.1%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if -4.999999999999985e-310 < d < 1.09999999999999994e-92

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 43.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 43.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 43.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}} \]

      4. associate-/l* 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125}{\frac{d}{{D}^{2} \cdot {M}^{2}}}} \]

      5. unpow2 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}} \]

      6. associate-*r* 48.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\color{blue}{\left({D}^{2} \cdot M\right) \cdot M}}} \]

      7. unpow2 48.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M}} \]

   4. Simplified 48.4%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}}} \]
   5. Step-by-step derivation

      1. sqrt-div 53.8%

         \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   6. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]
   7. Step-by-step derivation

      1. sqr-pow 53.8%

         \[\leadsto \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      2. rem-sqrt-square 56.4%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      3. sqr-pow 56.3%

         \[\leadsto \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      4. fabs-sqr 56.3%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      5. sqr-pow 56.4%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      6. metadata-eval 56.4%

         \[\leadsto \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   8. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   if 1.09999999999999994e-92 < d

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 47.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 47.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 47.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 47.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 47.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 47.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 47.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 47.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 47.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 47.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 64.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-52}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right)}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 10: 48.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-93}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5.5e-93)
   (* d (cbrt (* (/ -1.0 (* h l)) (sqrt (/ (/ 1.0 h) l)))))
   (if (<= d -5e-310)
     (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
     (if (<= d 7e-93)
       (* -0.125 (* (* D (* D (* M (/ M d)))) (sqrt (/ h (pow l 3.0)))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.5e-93) {
		tmp = d * cbrt(((-1.0 / (h * l)) * sqrt(((1.0 / h) / l))));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 7e-93) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * sqrt((h / pow(l, 3.0))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.5e-93) {
		tmp = d * Math.cbrt(((-1.0 / (h * l)) * Math.sqrt(((1.0 / h) / l))));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 7e-93) {
		tmp = -0.125 * ((D * (D * (M * (M / d)))) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5.5e-93)
		tmp = Float64(d * cbrt(Float64(Float64(-1.0 / Float64(h * l)) * sqrt(Float64(Float64(1.0 / h) / l)))));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 7e-93)
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(M * Float64(M / d)))) * sqrt(Float64(h / (l ^ 3.0)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5.5e-93], N[(d * N[Power[N[(N[(-1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-93], N[(-0.125 * N[(N[(D * N[(D * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.49999999999999968e-93

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 9.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 17.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 17.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 17.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 12.4%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 12.4%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. div-inv 12.4%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. frac-2neg 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \color{blue}{\frac{-1}{-\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{\color{blue}{-1}}{-\ell}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. mul-1-neg 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{-1 \cdot \ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      5. add-sqr-sqrt 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      6. sqrt-unprod 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      7. mul-1-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. mul-1-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      9. sqr-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      10. sqrt-unprod 0.0%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      11. add-sqr-sqrt 45.5%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   8. Applied egg-rr 45.5%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]
   9. Step-by-step derivation

      1. associate-*r/ 45.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h} \cdot -1}{\ell}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. associate-*l/ 45.5%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1 \cdot -1}{h}}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 45.5%

         \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{-1}}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. associate-/l/ 45.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   10. Simplified 45.5%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -5.49999999999999968e-93 < d < -4.999999999999985e-310

   1. Initial program 46.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 12.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 12.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 12.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 12.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 12.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 12.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 12.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 12.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 12.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 12.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.7%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 29.9%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 29.9%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 47.1%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if -4.999999999999985e-310 < d < 7e-93

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 54.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 54.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 57.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 57.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 57.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 54.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 54.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0 43.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 40.6%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 40.6%

         \[\leadsto -0.125 \cdot \left(\left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 40.6%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      4. associate-*l* 51.1%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      5. associate-/l* 56.2%

         \[\leadsto -0.125 \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      6. associate-/r/ 56.2%

         \[\leadsto -0.125 \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot M\right)}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   if 7e-93 < d

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 47.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 47.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 47.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 47.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 47.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 47.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 47.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 47.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 47.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 47.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 64.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-93}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 49.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.5e-93)
   (* d (cbrt (* (/ -1.0 (* h l)) (sqrt (/ (/ 1.0 h) l)))))
   (if (<= d -5e-310)
     (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
     (if (<= d 9.5e-93)
       (* (/ (sqrt h) (pow l 1.5)) (/ -0.125 (/ d (* M (* M (* D D))))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.5e-93) {
		tmp = d * cbrt(((-1.0 / (h * l)) * sqrt(((1.0 / h) / l))));
	} else if (d <= -5e-310) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 9.5e-93) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.5e-93) {
		tmp = d * Math.cbrt(((-1.0 / (h * l)) * Math.sqrt(((1.0 / h) / l))));
	} else if (d <= -5e-310) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (d <= 9.5e-93) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 / (d / (M * (M * (D * D)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.5e-93)
		tmp = Float64(d * cbrt(Float64(Float64(-1.0 / Float64(h * l)) * sqrt(Float64(Float64(1.0 / h) / l)))));
	elseif (d <= -5e-310)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (d <= 9.5e-93)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 / Float64(d / Float64(M * Float64(M * Float64(D * D))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.5e-93], N[(d * N[Power[N[(N[(-1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e-93], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d / N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.49999999999999997e-93

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 9.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 17.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 17.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 17.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 17.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 17.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 12.4%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 12.4%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. div-inv 12.4%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. frac-2neg 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \color{blue}{\frac{-1}{-\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{\color{blue}{-1}}{-\ell}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. mul-1-neg 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{-1 \cdot \ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      5. add-sqr-sqrt 12.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      6. sqrt-unprod 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      7. mul-1-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. mul-1-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      9. sqr-neg 21.0%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      10. sqrt-unprod 0.0%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      11. add-sqr-sqrt 45.5%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   8. Applied egg-rr 45.5%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]
   9. Step-by-step derivation

      1. associate-*r/ 45.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h} \cdot -1}{\ell}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. associate-*l/ 45.5%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1 \cdot -1}{h}}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 45.5%

         \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{-1}}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. associate-/l/ 45.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   10. Simplified 45.5%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -2.49999999999999997e-93 < d < -4.999999999999985e-310

   1. Initial program 46.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 12.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 12.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 12.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 12.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 12.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 12.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 12.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 12.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 12.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 12.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 12.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.7%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 29.9%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 29.9%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 47.1%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if -4.999999999999985e-310 < d < 9.5000000000000001e-93

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 43.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 43.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 43.0%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}} \]

      4. associate-/l* 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{-0.125}{\frac{d}{{D}^{2} \cdot {M}^{2}}}} \]

      5. unpow2 43.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}} \]

      6. associate-*r* 48.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\color{blue}{\left({D}^{2} \cdot M\right) \cdot M}}} \]

      7. unpow2 48.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M}} \]

   4. Simplified 48.4%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}}} \]
   5. Step-by-step derivation

      1. sqrt-div 53.8%

         \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   6. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]
   7. Step-by-step derivation

      1. sqr-pow 53.8%

         \[\leadsto \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      2. rem-sqrt-square 56.4%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      3. sqr-pow 56.3%

         \[\leadsto \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      4. fabs-sqr 56.3%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      5. sqr-pow 56.4%

         \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

      6. metadata-eval 56.4%

         \[\leadsto \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   8. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \frac{-0.125}{\frac{d}{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}} \]

   if 9.5000000000000001e-93 < d

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 47.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 47.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 47.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 47.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 47.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 47.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 47.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 47.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 47.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 47.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 47.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 64.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \frac{-0.125}{\frac{d}{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{+54}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.9e+54)
   (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
   (if (<= h -2e-310)
     (* d (cbrt (* (sqrt (/ (/ 1.0 h) l)) (* (/ 1.0 h) (/ -1.0 l)))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.9e+54) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (h <= -2e-310) {
		tmp = d * cbrt((sqrt(((1.0 / h) / l)) * ((1.0 / h) * (-1.0 / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.9e+54) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (h <= -2e-310) {
		tmp = d * Math.cbrt((Math.sqrt(((1.0 / h) / l)) * ((1.0 / h) * (-1.0 / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.9e+54)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (h <= -2e-310)
		tmp = Float64(d * cbrt(Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(Float64(1.0 / h) * Float64(-1.0 / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.9e+54], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(d * N[Power[N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / h), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9000000000000001e54

   1. Initial program 61.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 6.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 6.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 6.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 6.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 6.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 6.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 6.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 6.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 6.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 6.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 6.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 6.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.8%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 23.8%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 23.8%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 57.7%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if -1.9000000000000001e54 < h < -1.999999999999994e-310

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 13.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 23.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 23.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 23.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 18.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 18.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. div-inv 18.3%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. frac-2neg 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \color{blue}{\frac{-1}{-\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{\color{blue}{-1}}{-\ell}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. mul-1-neg 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{-1 \cdot \ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      5. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      7. mul-1-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. mul-1-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      9. sqr-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      10. sqrt-unprod 0.0%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      11. add-sqr-sqrt 45.8%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   8. Applied egg-rr 45.8%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -1.999999999999994e-310 < h

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{+54}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 13: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.6 \cdot 10^{+52}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.6e+52)
   (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
   (if (<= h -2e-310)
     (* d (cbrt (* (/ -1.0 (* h l)) (sqrt (/ (/ 1.0 h) l)))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.6e+52) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else if (h <= -2e-310) {
		tmp = d * cbrt(((-1.0 / (h * l)) * sqrt(((1.0 / h) / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.6e+52) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else if (h <= -2e-310) {
		tmp = d * Math.cbrt(((-1.0 / (h * l)) * Math.sqrt(((1.0 / h) / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.6e+52)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	elseif (h <= -2e-310)
		tmp = Float64(d * cbrt(Float64(Float64(-1.0 / Float64(h * l)) * sqrt(Float64(Float64(1.0 / h) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.6e+52], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(d * N[Power[N[(N[(-1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6e52

   1. Initial program 61.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 6.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 6.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 6.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 6.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 6.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 6.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 6.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 6.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 6.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 6.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 6.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 6.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.8%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 23.8%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 23.8%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 57.7%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if -1.6e52 < h < -1.999999999999994e-310

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 13.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 23.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 23.4%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 23.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 23.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 23.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 18.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 18.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. div-inv 18.3%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. frac-2neg 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \color{blue}{\frac{-1}{-\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{\color{blue}{-1}}{-\ell}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. mul-1-neg 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{-1 \cdot \ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      5. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      7. mul-1-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. mul-1-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      9. sqr-neg 18.4%

         \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      10. sqrt-unprod 0.0%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      11. add-sqr-sqrt 45.8%

          \[\leadsto \sqrt[3]{\left(\frac{1}{h} \cdot \frac{-1}{\color{blue}{\ell}}\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   8. Applied egg-rr 45.8%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{1}{h} \cdot \frac{-1}{\ell}\right)} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]
   9. Step-by-step derivation

      1. associate-*r/ 45.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h} \cdot -1}{\ell}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. associate-*l/ 45.8%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1 \cdot -1}{h}}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      3. metadata-eval 45.8%

         \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{-1}}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      4. associate-/l/ 45.7%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   10. Simplified 45.7%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{\ell \cdot h}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -1.999999999999994e-310 < h

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.6 \cdot 10^{+52}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{-1}{h \cdot \ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 47.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.32 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-303}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.32e-94)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 1.6e-303)
     (* d (exp (* -0.5 (log1p (+ (* l (/ 1.0 h)) -1.0)))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.32e-94) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 1.6e-303) {
		tmp = d * exp((-0.5 * log1p(((l * (1.0 / h)) + -1.0))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.32e-94) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 1.6e-303) {
		tmp = d * Math.exp((-0.5 * Math.log1p(((l * (1.0 / h)) + -1.0))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.32e-94:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 1.6e-303:
		tmp = d * math.exp((-0.5 * math.log1p(((l * (1.0 / h)) + -1.0))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.32e-94)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 1.6e-303)
		tmp = Float64(d * exp(Float64(-0.5 * log1p(Float64(Float64(l * Float64(1.0 / h)) + -1.0)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.32e-94], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e-303], N[(d * N[Exp[N[(-0.5 * N[Log[1 + N[(N[(l * N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.32000000000000002e-94

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 71.4%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 71.4%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 71.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 71.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 71.4%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 43.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -2.32000000000000002e-94 < d < 1.59999999999999995e-303

   1. Initial program 45.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 12.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 12.5%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 12.5%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 12.5%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 12.5%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 12.5%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 12.5%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 12.5%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 12.5%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 12.5%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 12.5%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 12.5%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 6.4%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. log1p-expm1-u 28.6%

         \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)\right)}} \cdot d \]

      2. expm1-udef 28.6%

         \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)} - 1}\right)} \cdot d \]

   9. Applied egg-rr 45.3%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\ell \cdot \frac{1}{h} - 1\right)}} \cdot d \]

   if 1.59999999999999995e-303 < d

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.6%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.6%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.32 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-303}:\\ \;\;\;\;d \cdot e^{-0.5 \cdot \mathsf{log1p}\left(\ell \cdot \frac{1}{h} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 15: 34.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -6.4e-15)
   (* d (pow (/ l h) -0.5))
   (if (<= h -2e-310)
     (* d (cbrt (pow (/ 1.0 (* h l)) 1.5)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * cbrt(pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * Math.pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -6.4e-15)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	elseif (h <= -2e-310)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(h * l)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -6.4e-15], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(d * N[Power[N[Power[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.3999999999999999e-15

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -6.3999999999999999e-15 < h < -1.999999999999994e-310

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 14.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 27.1%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 27.1%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 27.1%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 27.1%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 27.1%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 27.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 27.1%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 27.1%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 27.1%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 20.9%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 20.9%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. expm1-log1p-u 20.9%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right)}} \cdot d \]

      2. expm1-udef 21.0%

         \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} - 1}} \cdot d \]

      3. pow1 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} - 1} \cdot d \]

      4. pow1/2 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}}\right)} - 1} \cdot d \]

      5. pow-prod-up 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1 + 0.5\right)}}\right)} - 1} \cdot d \]

      6. associate-/l/ 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{\left(1 + 0.5\right)}\right)} - 1} \cdot d \]

      7. inv-pow 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{\left(1 + 0.5\right)}\right)} - 1} \cdot d \]

      8. metadata-eval 21.0%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\color{blue}{1.5}}\right)} - 1} \cdot d \]

   8. Applied egg-rr 21.0%

      \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}\right)} - 1}} \cdot d \]
   9. Step-by-step derivation

      1. expm1-def 20.9%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}\right)\right)}} \cdot d \]

      2. expm1-log1p 20.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}}} \cdot d \]

      3. unpow-1 20.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{1.5}} \cdot d \]

   10. Simplified 20.9%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \cdot d \]

   if -1.999999999999994e-310 < h

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 16: 45.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9e-177)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l -5e-311)
     (* d (cbrt (pow (/ 1.0 (* h l)) 1.5)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e-177) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -5e-311) {
		tmp = d * cbrt(pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e-177) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -5e-311) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9e-177)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -5e-311)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(h * l)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9e-177], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(d * N[Power[N[Power[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.0000000000000007e-177

   1. Initial program 57.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 57.6%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 57.6%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 57.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 57.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 57.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 38.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -9.0000000000000007e-177 < l < -5.00000000000023e-311

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 31.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 51.5%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

      2. *-commutative 51.5%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{\color{blue}{h \cdot \ell}} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      3. *-commutative 51.5%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{\color{blue}{h \cdot \ell}}\right) \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      4. *-commutative 51.5%

         \[\leadsto \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{\color{blue}{h \cdot \ell}}}} \cdot d \]

   4. Applied egg-rr 51.5%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]
   5. Step-by-step derivation

      1. add-cbrt-cube 51.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}\right) \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}}} \cdot d \]

      2. add-sqr-sqrt 51.5%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      3. add-cbrt-cube 51.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      4. associate-/r* 51.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \cdot d \]

      5. add-cbrt-cube 44.9%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      6. associate-/r* 44.9%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   6. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   7. Step-by-step derivation

      1. expm1-log1p-u 44.9%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right)}} \cdot d \]

      2. expm1-udef 44.7%

         \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} - 1}} \cdot d \]

      3. pow1 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} - 1} \cdot d \]

      4. pow1/2 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}}\right)} - 1} \cdot d \]

      5. pow-prod-up 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1 + 0.5\right)}}\right)} - 1} \cdot d \]

      6. associate-/l/ 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{\left(1 + 0.5\right)}\right)} - 1} \cdot d \]

      7. inv-pow 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{\left(1 + 0.5\right)}\right)} - 1} \cdot d \]

      8. metadata-eval 44.7%

         \[\leadsto \sqrt[3]{e^{\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\color{blue}{1.5}}\right)} - 1} \cdot d \]

   8. Applied egg-rr 44.7%

      \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}\right)} - 1}} \cdot d \]
   9. Step-by-step derivation

      1. expm1-def 44.9%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}\right)\right)}} \cdot d \]

      2. expm1-log1p 44.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{1.5}}} \cdot d \]

      3. unpow-1 44.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{1.5}} \cdot d \]

   10. Simplified 44.9%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \cdot d \]

   if -5.00000000000023e-311 < l

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17: 33.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -6.4e-15)
   (* d (pow (/ l h) -0.5))
   (if (<= h -2e-310)
     (* d (sqrt (/ (/ 1.0 h) l)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-6.4d-15)) then
        tmp = d * ((l / h) ** (-0.5d0))
    else if (h <= (-2d-310)) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * Math.pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -6.4e-15:
		tmp = d * math.pow((l / h), -0.5)
	elif h <= -2e-310:
		tmp = d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -6.4e-15)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	elseif (h <= -2e-310)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -6.4e-15)
		tmp = d * ((l / h) ^ -0.5);
	elseif (h <= -2e-310)
		tmp = d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -6.4e-15], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.3999999999999999e-15

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -6.3999999999999999e-15 < h < -1.999999999999994e-310

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 14.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 14.7%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      2. *-commutative 14.7%

         \[\leadsto \sqrt{1 \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   4. Applied egg-rr 14.7%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 14.7%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. associate-/r* 14.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   6. Simplified 14.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -1.999999999999994e-310 < h

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   7. Step-by-step derivation

      1. unpow-prod-down 51.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 18: 33.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.9e-15)
   (* d (pow (/ l h) -0.5))
   (if (<= h -2e-310)
     (* d (sqrt (/ (/ 1.0 h) l)))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.9e-15) {
		tmp = d * pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1.9d-15)) then
        tmp = d * ((l / h) ** (-0.5d0))
    else if (h <= (-2d-310)) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.9e-15) {
		tmp = d * Math.pow((l / h), -0.5);
	} else if (h <= -2e-310) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.9e-15:
		tmp = d * math.pow((l / h), -0.5)
	elif h <= -2e-310:
		tmp = d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.9e-15)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	elseif (h <= -2e-310)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.9e-15)
		tmp = d * ((l / h) ^ -0.5);
	elseif (h <= -2e-310)
		tmp = d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.9e-15], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9000000000000001e-15

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -1.9000000000000001e-15 < h < -1.999999999999994e-310

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 14.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 14.7%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      2. *-commutative 14.7%

         \[\leadsto \sqrt{1 \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   4. Applied egg-rr 14.7%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 14.7%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. associate-/r* 14.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   6. Simplified 14.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   if -1.999999999999994e-310 < h

   1. Initial program 73.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 38.2%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      2. *-commutative 38.2%

         \[\leadsto \sqrt{1 \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   4. Applied egg-rr 38.2%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 38.2%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. associate-/r* 40.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   6. Simplified 40.1%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]
   7. Step-by-step derivation

      1. sqrt-div 51.0%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \cdot d \]

      2. inv-pow 51.0%

         \[\leadsto \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \cdot d \]

      3. sqrt-pow1 51.0%

         \[\leadsto \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \cdot d \]

      4. metadata-eval 51.0%

         \[\leadsto \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \cdot d \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 19: 29.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -4.6e-17) (* d (pow (/ l h) -0.5)) (* d (sqrt (/ 1.0 (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -4.6e-17) {
		tmp = d * pow((l / h), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-4.6d-17)) then
        tmp = d * ((l / h) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -4.6e-17) {
		tmp = d * Math.pow((l / h), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -4.6e-17:
		tmp = d * math.pow((l / h), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -4.6e-17)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -4.6e-17)
		tmp = d * ((l / h) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -4.6e-17], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -4.60000000000000018e-17

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -4.60000000000000018e-17 < h

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 20: 29.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -3.4e-15) (* d (pow (/ l h) -0.5)) (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -3.4e-15) {
		tmp = d * pow((l / h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-3.4d-15)) then
        tmp = d * ((l / h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -3.4e-15) {
		tmp = d * Math.pow((l / h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -3.4e-15:
		tmp = d * math.pow((l / h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -3.4e-15)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -3.4e-15)
		tmp = d * ((l / h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -3.4e-15], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -3.4e-15

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -3.4e-15 < h

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 30.6%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{\ell \cdot h}}} \cdot d \]

      2. *-commutative 30.6%

         \[\leadsto \sqrt{1 \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   4. Applied egg-rr 30.6%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 30.6%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. associate-/r* 31.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   6. Simplified 31.9%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 21: 29.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -6.4e-15) (* d (pow (/ l h) -0.5)) (* d (pow (* h l) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * pow((l / h), -0.5);
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-6.4d-15)) then
        tmp = d * ((l / h) ** (-0.5d0))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6.4e-15) {
		tmp = d * Math.pow((l / h), -0.5);
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -6.4e-15:
		tmp = d * math.pow((l / h), -0.5)
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -6.4e-15)
		tmp = Float64(d * (Float64(l / h) ^ -0.5));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -6.4e-15)
		tmp = d * ((l / h) ^ -0.5);
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -6.4e-15], N[(d * N[Power[N[(l / h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -6.3999999999999999e-15

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 7.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 7.1%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 7.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 7.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 7.1%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 7.1%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   7. Taylor expanded in h around -inf 7.0%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}} \cdot d \]
   8. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right)} \cdot d \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)} - 1\right)} \cdot d \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)} - 1\right)} \cdot d \]
   10. Step-by-step derivation

       1. expm1-def 28.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}\right)\right)} \cdot d \]

       2. expm1-log1p 28.0%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{1}{h}\right)}^{-0.5}} \cdot d \]

       3. associate-*r/ 28.0%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{h}\right)}}^{-0.5} \cdot d \]

       4. *-rgt-identity 28.0%

          \[\leadsto {\left(\frac{\color{blue}{\ell}}{h}\right)}^{-0.5} \cdot d \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{h}\right)}^{-0.5}} \cdot d \]

   if -6.3999999999999999e-15 < h

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 30.6%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 30.6%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   4. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   5. Step-by-step derivation

      1. *-lft-identity 30.6%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 30.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 30.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 30.6%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 30.5%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 30.5%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 30.6%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 30.6%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   6. Simplified 30.6%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;d \cdot {\left(\frac{\ell}{h}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 22: 26.7% 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 67.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf 25.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Step-by-step derivation

   1. *-un-lft-identity 25.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 25.0%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

4. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
5. Step-by-step derivation

   1. *-lft-identity 25.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. unpow-1 25.0%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   3. sqr-pow 25.0%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   4. rem-sqrt-square 25.0%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   5. sqr-pow 25.0%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   6. fabs-sqr 25.0%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   7. sqr-pow 25.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   8. metadata-eval 25.0%

      \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

6. Simplified 25.0%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

7. Final simplification 25.0%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023181 
(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)))))