Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.6% → 79.4%

Time: 23.8s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(-d\right)}^{0.25} \cdot {\left(\frac{-1}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5e-311)
   (*
    (* (sqrt (/ d h)) (pow (* (pow (- d) 0.25) (pow (/ -1.0 l) 0.25)) 2.0))
    (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l))))
   (*
    (/ (sqrt d) (sqrt l))
    (*
     (/ (sqrt d) (sqrt h))
     (fma (pow (* 0.5 (/ M (/ d D))) 2.0) (* -0.5 (/ h l)) 1.0)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5e-311) {
		tmp = (sqrt((d / h)) * pow((pow(-d, 0.25) * pow((-1.0 / l), 0.25)), 2.0)) * (1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * fma(pow((0.5 * (M / (d / D))), 2.0), (-0.5 * (h / l)), 1.0));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5e-311)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * (Float64((Float64(-d) ^ 0.25) * (Float64(-1.0 / l) ^ 0.25)) ^ 2.0)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * fma((Float64(0.5 * Float64(M / Float64(d / D))) ^ 2.0), Float64(-0.5 * Float64(h / l)), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5e-311], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[Power[(-d), 0.25], $MachinePrecision] * N[Power[N[(-1.0 / l), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(0.5 * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000023e-311

   1. Initial program 70.9%

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

   3. Simplified 69.2%

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

      1. associate-*r/ 72.5%

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

      2. div-inv 72.5%

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

      3. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. pow1/2 72.5%

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

      2. sqr-pow 72.3%

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

      3. pow2 72.3%

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

      4. metadata-eval 72.3%

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

   7. Applied egg-rr 72.3%

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

   8. Taylor expanded in l around -inf 73.7%

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

      1. distribute-lft-in 73.7%

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

      2. exp-sum 73.9%

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

      3. *-commutative 73.9%

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

      4. rem-square-sqrt 0.0%

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

      5. unpow2 0.0%

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

      6. *-commutative 0.0%

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

      7. exp-to-pow 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 74.2%

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

      11. mul-1-neg 74.2%

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

      12. *-commutative 74.2%

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

      13. exp-to-pow 76.7%

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

   10. Simplified 76.7%

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

   if -5.00000000000023e-311 < l

   1. Initial program 67.2%

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

   3. Simplified 67.3%

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

      1. sqrt-div 80.0%

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

   5. Applied egg-rr 80.0%

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

      1. sqrt-div 87.8%

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

   7. Applied egg-rr 87.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(-d\right)}^{0.25} \cdot {\left(\frac{-1}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \end{array} \]

Alternative 2: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(-d\right)}^{0.25} \cdot {\left(\frac{-1}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l)))))
   (if (<= d -1.95e-296)
     (*
      (* (sqrt (/ d h)) (pow (* (pow (- d) 0.25) (pow (/ -1.0 l) 0.25)) 2.0))
      t_0)
     (if (<= d 3.4e-184)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (* t_0 (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = (sqrt((d / h)) * pow((pow(-d, 0.25) * pow((-1.0 / l), 0.25)), 2.0)) * t_0;
	} else if (d <= 3.4e-184) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else {
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * sqrt((d / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0)) / l))
    if (d <= (-1.95d-296)) then
        tmp = (sqrt((d / h)) * (((-d ** 0.25d0) * (((-1.0d0) / l) ** 0.25d0)) ** 2.0d0)) * t_0
    else if (d <= 3.4d-184) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else
        tmp = t_0 * ((sqrt(d) / sqrt(h)) * sqrt((d / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * Math.pow(((0.5 * M) * (D / d)), 2.0)) / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = (Math.sqrt((d / h)) * Math.pow((Math.pow(-d, 0.25) * Math.pow((-1.0 / l), 0.25)), 2.0)) * t_0;
	} else if (d <= 3.4e-184) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * ((h * math.pow(((0.5 * M) * (D / d)), 2.0)) / l))
	tmp = 0
	if d <= -1.95e-296:
		tmp = (math.sqrt((d / h)) * math.pow((math.pow(-d, 0.25) * math.pow((-1.0 / l), 0.25)), 2.0)) * t_0
	elif d <= 3.4e-184:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	else:
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l)))
	tmp = 0.0
	if (d <= -1.95e-296)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * (Float64((Float64(-d) ^ 0.25) * (Float64(-1.0 / l) ^ 0.25)) ^ 2.0)) * t_0);
	elseif (d <= 3.4e-184)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	else
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * ((h * (((0.5 * M) * (D / d)) ^ 2.0)) / l));
	tmp = 0.0;
	if (d <= -1.95e-296)
		tmp = (sqrt((d / h)) * (((-d ^ 0.25) * ((-1.0 / l) ^ 0.25)) ^ 2.0)) * t_0;
	elseif (d <= 3.4e-184)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	else
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.95e-296], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[Power[(-d), 0.25], $MachinePrecision] * N[Power[N[(-1.0 / l), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 3.4e-184], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.95000000000000005e-296

   1. Initial program 72.1%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. metadata-eval 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. pow1/2 73.7%

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

      2. sqr-pow 73.6%

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

      3. pow2 73.6%

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

      4. metadata-eval 73.6%

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

   7. Applied egg-rr 73.6%

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

   8. Taylor expanded in l around -inf 74.9%

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

      1. distribute-lft-in 74.9%

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

      2. exp-sum 75.1%

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

      3. *-commutative 75.1%

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

      4. rem-square-sqrt 0.0%

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

      5. unpow2 0.0%

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

      6. *-commutative 0.0%

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

      7. exp-to-pow 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 75.4%

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

      11. mul-1-neg 75.4%

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

      12. *-commutative 75.4%

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

      13. exp-to-pow 78.0%

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

   10. Simplified 78.0%

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

   if -1.95000000000000005e-296 < d < 3.40000000000000004e-184

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

      1. associate-*r/ 34.9%

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

      2. div-inv 34.9%

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

      3. metadata-eval 34.9%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in d around 0 43.2%

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

      1. associate-*l/ 43.3%

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

      2. unpow2 43.3%

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

      3. unpow2 43.3%

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

      4. unswap-sqr 55.9%

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

   8. Simplified 55.9%

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

      1. sqrt-div 55.9%

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

   10. Applied egg-rr 55.9%

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

       1. sqr-pow 55.9%

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

       2. rem-sqrt-square 63.6%

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

       3. sqr-pow 63.6%

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

       4. fabs-sqr 63.6%

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

       5. sqr-pow 63.6%

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

       6. metadata-eval 63.6%

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

   12. Simplified 63.6%

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

   if 3.40000000000000004e-184 < d

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r/ 77.9%

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

      2. div-inv 77.9%

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

      3. metadata-eval 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. sqrt-div 86.4%

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

   7. Applied egg-rr 88.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(-d\right)}^{0.25} \cdot {\left(\frac{-1}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]

Alternative 3: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot t_0\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -1.95e-296)
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l)))
      (* (sqrt (/ d h)) t_0))
     (if (<= d 1.3e-185)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (*
        (* (/ (sqrt d) (sqrt h)) t_0)
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = (1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (sqrt((d / h)) * t_0);
	} else if (d <= 1.3e-185) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (d <= (-1.95d-296)) then
        tmp = (1.0d0 - (0.5d0 * ((h * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0)) / l))) * (sqrt((d / h)) * t_0)
    else if (d <= 1.3d-185) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_0) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = (1.0 - (0.5 * ((h * Math.pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (Math.sqrt((d / h)) * t_0);
	} else if (d <= 1.3e-185) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_0) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.95e-296:
		tmp = (1.0 - (0.5 * ((h * math.pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (math.sqrt((d / h)) * t_0)
	elif d <= 1.3e-185:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_0) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.95e-296)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / h)) * t_0));
	elseif (d <= 1.3e-185)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.95e-296)
		tmp = (1.0 - (0.5 * ((h * (((0.5 * M) * (D / d)) ^ 2.0)) / l))) * (sqrt((d / h)) * t_0);
	elseif (d <= 1.3e-185)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e-296], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-185], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.95000000000000005e-296

   1. Initial program 72.1%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. metadata-eval 73.7%

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

   5. Applied egg-rr 73.7%

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

   if -1.95000000000000005e-296 < d < 1.29999999999999992e-185

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

      1. associate-*r/ 34.9%

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

      2. div-inv 34.9%

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

      3. metadata-eval 34.9%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in d around 0 43.2%

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

      1. associate-*l/ 43.3%

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

      2. unpow2 43.3%

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

      3. unpow2 43.3%

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

      4. unswap-sqr 55.9%

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

   8. Simplified 55.9%

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

      1. sqrt-div 55.9%

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

   10. Applied egg-rr 55.9%

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

       1. sqr-pow 55.9%

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

       2. rem-sqrt-square 63.6%

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

       3. sqr-pow 63.6%

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

       4. fabs-sqr 63.6%

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

       5. sqr-pow 63.6%

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

       6. metadata-eval 63.6%

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

   12. Simplified 63.6%

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

   if 1.29999999999999992e-185 < d

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. sqrt-div 86.4%

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

   5. Applied egg-rr 86.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 4: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot t_1\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-187}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l))))
        (t_1 (sqrt (/ d l))))
   (if (<= d -1.95e-296)
     (* t_0 (* (sqrt (/ d h)) t_1))
     (if (<= d 3.2e-187)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (* t_0 (* (/ (sqrt d) (sqrt h)) t_1))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l));
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = t_0 * (sqrt((d / h)) * t_1);
	} else if (d <= 3.2e-187) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else {
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * t_1);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0)) / l))
    t_1 = sqrt((d / l))
    if (d <= (-1.95d-296)) then
        tmp = t_0 * (sqrt((d / h)) * t_1)
    else if (d <= 3.2d-187) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else
        tmp = t_0 * ((sqrt(d) / sqrt(h)) * t_1)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * Math.pow(((0.5 * M) * (D / d)), 2.0)) / l));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.95e-296) {
		tmp = t_0 * (Math.sqrt((d / h)) * t_1);
	} else if (d <= 3.2e-187) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(h)) * t_1);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * ((h * math.pow(((0.5 * M) * (D / d)), 2.0)) / l))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.95e-296:
		tmp = t_0 * (math.sqrt((d / h)) * t_1)
	elif d <= 3.2e-187:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	else:
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(h)) * t_1)
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l)))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.95e-296)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (d <= 3.2e-187)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	else
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(h)) * t_1));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * ((h * (((0.5 * M) * (D / d)) ^ 2.0)) / l));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.95e-296)
		tmp = t_0 * (sqrt((d / h)) * t_1);
	elseif (d <= 3.2e-187)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	else
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e-296], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-187], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.95000000000000005e-296

   1. Initial program 72.1%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. metadata-eval 73.7%

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

   5. Applied egg-rr 73.7%

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

   if -1.95000000000000005e-296 < d < 3.1999999999999998e-187

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

      1. associate-*r/ 34.9%

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

      2. div-inv 34.9%

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

      3. metadata-eval 34.9%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in d around 0 43.2%

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

      1. associate-*l/ 43.3%

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

      2. unpow2 43.3%

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

      3. unpow2 43.3%

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

      4. unswap-sqr 55.9%

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

   8. Simplified 55.9%

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

      1. sqrt-div 55.9%

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

   10. Applied egg-rr 55.9%

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

       1. sqr-pow 55.9%

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

       2. rem-sqrt-square 63.6%

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

       3. sqr-pow 63.6%

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

       4. fabs-sqr 63.6%

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

       5. sqr-pow 63.6%

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

       6. metadata-eval 63.6%

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

   12. Simplified 63.6%

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

   if 3.1999999999999998e-187 < d

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r/ 77.9%

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

      2. div-inv 77.9%

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

      3. metadata-eval 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. sqrt-div 86.4%

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

   7. Applied egg-rr 88.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-187}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]

Alternative 5: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 5.5e+118)
   (*
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5.5e+118) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5.5d+118) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5.5e+118) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 5.5e+118:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 5.5e+118)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 5.5e+118)
		tmp = (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 5.5e+118], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5000000000000003e118

   1. Initial program 72.0%

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

   3. Simplified 71.1%

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

   if 5.5000000000000003e118 < l

   1. Initial program 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in d around inf 50.2%

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

      1. associate-/r* 52.1%

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

   6. Simplified 52.1%

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

      1. sqrt-div 75.5%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+120}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.5e+120)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l)))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e+120) {
		tmp = (1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.5d+120) then
        tmp = (1.0d0 - (0.5d0 * ((h * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0)) / l))) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e+120) {
		tmp = (1.0 - (0.5 * ((h * Math.pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.5e+120:
		tmp = (1.0 - (0.5 * ((h * math.pow(((0.5 * M) * (D / d)), 2.0)) / l))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.5e+120)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.5e+120)
		tmp = (1.0 - (0.5 * ((h * (((0.5 * M) * (D / d)) ^ 2.0)) / l))) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.5e+120], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000009e120

   1. Initial program 72.0%

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

   3. Simplified 71.1%

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

      1. associate-*r/ 73.9%

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

      2. div-inv 73.9%

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

      3. metadata-eval 73.9%

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

   5. Applied egg-rr 73.9%

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

   if 2.50000000000000009e120 < l

   1. Initial program 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in d around inf 50.2%

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

      1. associate-/r* 52.1%

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

   6. Simplified 52.1%

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

      1. sqrt-div 75.5%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+120}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-58}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -3.8e-196)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* 0.5 (* M (/ D d))) 2.0)))))
   (if (<= d 4.5e-305)
     (* d (log (exp (pow (* l h) -0.5))))
     (if (<= d 6e-58)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (if (<= d 4.5e+87)
         (*
          (sqrt (/ (* d (/ d h)) l))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* (* 0.5 M) (/ D d)) 2.0)))))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.8e-196) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 4.5e-305) {
		tmp = d * log(exp(pow((l * h), -0.5)));
	} else if (d <= 6e-58) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else if (d <= 4.5e+87) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= (-3.8d-196)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    else if (d <= 4.5d-305) then
        tmp = d * log(exp(((l * h) ** (-0.5d0))))
    else if (d <= 6d-58) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else if (d <= 4.5d+87) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * ((h / l) * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.8e-196) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 4.5e-305) {
		tmp = d * Math.log(Math.exp(Math.pow((l * h), -0.5)));
	} else if (d <= 6e-58) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else if (d <= 4.5e+87) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * Math.pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -3.8e-196:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((0.5 * (M * (D / d))), 2.0))))
	elif d <= 4.5e-305:
		tmp = d * math.log(math.exp(math.pow((l * h), -0.5)))
	elif d <= 6e-58:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	elif d <= 4.5e+87:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * math.pow(((0.5 * M) * (D / d)), 2.0))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.8e-196)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	elseif (d <= 4.5e-305)
		tmp = Float64(d * log(exp((Float64(l * h) ^ -0.5))));
	elseif (d <= 6e-58)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	elseif (d <= 4.5e+87)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -3.8e-196)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((0.5 * (M * (D / d))) ^ 2.0))));
	elseif (d <= 4.5e-305)
		tmp = d * log(exp(((l * h) ^ -0.5)));
	elseif (d <= 6e-58)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	elseif (d <= 4.5e+87)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * (((0.5 * M) * (D / d)) ^ 2.0))));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -3.8e-196], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-305], N[(d * N[Log[N[Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6e-58], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+87], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.8000000000000001e-196

   1. Initial program 76.7%

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

   3. Simplified 74.4%

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

      1. associate-*r/ 78.8%

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

      2. div-inv 78.8%

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

      3. metadata-eval 78.8%

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

   5. Applied egg-rr 78.8%

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

   7. Applied egg-rr 65.4%

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

      1. unpow1 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   9. Simplified 65.4%

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

   if -3.8000000000000001e-196 < d < 4.5000000000000002e-305

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

      1. associate-*r/ 51.6%

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

      2. div-inv 51.6%

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

      3. metadata-eval 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in d around inf 24.3%

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

      1. unpow-1 24.3%

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

      2. sqr-pow 24.3%

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

      3. rem-sqrt-square 21.2%

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

      4. metadata-eval 21.2%

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

      5. sqr-pow 21.2%

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

      6. fabs-sqr 21.2%

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

      7. sqr-pow 21.2%

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

   8. Simplified 21.2%

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

      1. add-log-exp 49.6%

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

      2. *-commutative 49.6%

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

   10. Applied egg-rr 49.6%

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

   if 4.5000000000000002e-305 < d < 6.00000000000000015e-58

   1. Initial program 48.6%

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

   3. Simplified 48.4%

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

      1. associate-*r/ 46.7%

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

      2. div-inv 46.7%

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

      3. metadata-eval 46.7%

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

   5. Applied egg-rr 46.7%

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

   6. Taylor expanded in d around 0 40.1%

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

      1. associate-*l/ 40.1%

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

      2. unpow2 40.1%

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

      3. unpow2 40.1%

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

      4. unswap-sqr 54.1%

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

   8. Simplified 54.1%

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

      1. sqrt-div 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. sqr-pow 56.1%

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

       2. rem-sqrt-square 64.9%

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

       3. sqr-pow 64.9%

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

       4. fabs-sqr 64.9%

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

       5. sqr-pow 64.9%

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

       6. metadata-eval 64.9%

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

   12. Simplified 64.9%

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

   if 6.00000000000000015e-58 < d < 4.5000000000000003e87

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Applied egg-rr 15.7%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 70.0%

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

      3. associate-*r/ 70.1%

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

      4. *-commutative 70.1%

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

   7. Simplified 70.1%

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

   if 4.5000000000000003e87 < d

   1. Initial program 78.4%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in d around inf 68.9%

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

      1. associate-/r* 68.9%

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

   6. Simplified 68.9%

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

      1. sqrt-div 86.6%

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

   8. Applied egg-rr 86.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-58}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.5e-196)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* 0.5 (* M (/ D d))) 2.0)))))
   (if (<= d -1.95e-296)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (* -0.125 (* (* D (/ D d)) (* (/ M d) (* M (/ h l)))))))
     (if (<= d 8.5e-60)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (if (<= d 2.5e+87)
         (*
          (sqrt (/ (* d (/ d h)) l))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* (* 0.5 M) (/ D d)) 2.0)))))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-196) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= -1.95e-296) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (-0.125 * ((D * (D / d)) * ((M / d) * (M * (h / l))))));
	} else if (d <= 8.5e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else if (d <= 2.5e+87) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= (-6.5d-196)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    else if (d <= (-1.95d-296)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * ((-0.125d0) * ((d_1 * (d_1 / d)) * ((m / d) * (m * (h / l))))))
    else if (d <= 8.5d-60) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else if (d <= 2.5d+87) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * ((h / l) * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-196) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= -1.95e-296) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (-0.125 * ((D * (D / d)) * ((M / d) * (M * (h / l))))));
	} else if (d <= 8.5e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else if (d <= 2.5e+87) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * Math.pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -6.5e-196:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((0.5 * (M * (D / d))), 2.0))))
	elif d <= -1.95e-296:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (-0.125 * ((D * (D / d)) * ((M / d) * (M * (h / l))))))
	elif d <= 8.5e-60:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	elif d <= 2.5e+87:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * math.pow(((0.5 * M) * (D / d)), 2.0))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.5e-196)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	elseif (d <= -1.95e-296)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(-0.125 * Float64(Float64(D * Float64(D / d)) * Float64(Float64(M / d) * Float64(M * Float64(h / l)))))));
	elseif (d <= 8.5e-60)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	elseif (d <= 2.5e+87)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -6.5e-196)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((0.5 * (M * (D / d))) ^ 2.0))));
	elseif (d <= -1.95e-296)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (-0.125 * ((D * (D / d)) * ((M / d) * (M * (h / l))))));
	elseif (d <= 8.5e-60)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	elseif (d <= 2.5e+87)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * (((0.5 * M) * (D / d)) ^ 2.0))));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.5e-196], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.95e-296], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e-60], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e+87], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.5000000000000004e-196

   1. Initial program 76.7%

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

   3. Simplified 74.4%

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

      1. associate-*r/ 78.8%

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

      2. div-inv 78.8%

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

      3. metadata-eval 78.8%

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

   5. Applied egg-rr 78.8%

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

   7. Applied egg-rr 65.4%

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

      1. unpow1 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   9. Simplified 65.4%

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

   if -6.5000000000000004e-196 < d < -1.95000000000000005e-296

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in M around inf 32.1%

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

      1. times-frac 28.6%

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

      2. unpow2 28.6%

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

      3. unpow2 28.6%

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

      4. unpow2 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in D around 0 32.1%

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

      1. *-commutative 32.1%

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

      2. unpow2 32.1%

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

      3. unpow2 32.1%

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

      4. associate-*l* 32.3%

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

      5. times-frac 39.5%

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

      6. unpow2 39.5%

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

      7. associate-*r/ 46.6%

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

      8. associate-*r* 46.6%

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

      9. *-commutative 46.6%

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

      10. *-commutative 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in M around 0 46.6%

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

       1. unpow2 46.6%

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

       2. associate-*r* 46.6%

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

       3. *-commutative 46.6%

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

       4. times-frac 50.3%

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

       5. associate-/l* 53.9%

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

       6. associate-/r/ 53.9%

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

   12. Simplified 53.9%

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

   if -1.95000000000000005e-296 < d < 8.50000000000000044e-60

   1. Initial program 45.9%

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

   3. Simplified 45.7%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around 0 37.8%

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

      1. associate-*l/ 37.8%

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

      2. unpow2 37.8%

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

      3. unpow2 37.8%

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

      4. unswap-sqr 51.0%

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

   8. Simplified 51.0%

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

      1. sqrt-div 52.9%

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

   10. Applied egg-rr 52.9%

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

       1. sqr-pow 52.9%

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

       2. rem-sqrt-square 61.3%

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

       3. sqr-pow 61.2%

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

       4. fabs-sqr 61.2%

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

       5. sqr-pow 61.3%

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

       6. metadata-eval 61.3%

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

   12. Simplified 61.3%

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

   if 8.50000000000000044e-60 < d < 2.4999999999999999e87

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Applied egg-rr 15.7%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 70.0%

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

      3. associate-*r/ 70.1%

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

      4. *-commutative 70.1%

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

   7. Simplified 70.1%

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

   if 2.4999999999999999e87 < d

   1. Initial program 78.4%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in d around inf 68.9%

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

      1. associate-/r* 68.9%

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

   6. Simplified 68.9%

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

      1. sqrt-div 86.6%

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

   8. Applied egg-rr 86.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-280}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d h) (/ d l)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* 0.5 (* M (/ D d))) 2.0)))))))
   (if (<= d -2.05e-280)
     t_0
     (if (<= d 8.5e-60)
       (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
       (if (<= d 3.6e+87) t_0 (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((0.5 * (M * (D / d))), 2.0))));
	double tmp;
	if (d <= -2.05e-280) {
		tmp = t_0;
	} else if (d <= 8.5e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else if (d <= 3.6e+87) {
		tmp = t_0;
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    if (d <= (-2.05d-280)) then
        tmp = t_0
    else if (d <= 8.5d-60) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else if (d <= 3.6d+87) then
        tmp = t_0
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((0.5 * (M * (D / d))), 2.0))));
	double tmp;
	if (d <= -2.05e-280) {
		tmp = t_0;
	} else if (d <= 8.5e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else if (d <= 3.6e+87) {
		tmp = t_0;
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((0.5 * (M * (D / d))), 2.0))))
	tmp = 0
	if d <= -2.05e-280:
		tmp = t_0
	elif d <= 8.5e-60:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	elif d <= 3.6e+87:
		tmp = t_0
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))))
	tmp = 0.0
	if (d <= -2.05e-280)
		tmp = t_0;
	elseif (d <= 8.5e-60)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	elseif (d <= 3.6e+87)
		tmp = t_0;
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((0.5 * (M * (D / d))) ^ 2.0))));
	tmp = 0.0;
	if (d <= -2.05e-280)
		tmp = t_0;
	elseif (d <= 8.5e-60)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	elseif (d <= 3.6e+87)
		tmp = t_0;
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.05e-280], t$95$0, If[LessEqual[d, 8.5e-60], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e+87], t$95$0, N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.0500000000000001e-280 or 8.50000000000000044e-60 < d < 3.59999999999999994e87

   1. Initial program 75.6%

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

   3. Simplified 74.2%

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

      1. associate-*r/ 77.6%

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

      2. div-inv 77.6%

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

      3. metadata-eval 77.6%

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

   5. Applied egg-rr 77.6%

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

   7. Applied egg-rr 63.5%

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

      1. unpow1 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*r* 63.5%

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

   9. Simplified 63.5%

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

   if -2.0500000000000001e-280 < d < 8.50000000000000044e-60

   1. Initial program 43.6%

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

   3. Simplified 43.5%

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

      1. associate-*r/ 42.0%

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

      2. div-inv 42.0%

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

      3. metadata-eval 42.0%

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

   5. Applied egg-rr 42.0%

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

   6. Taylor expanded in d around 0 34.6%

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

      1. associate-*l/ 34.7%

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

      2. unpow2 34.7%

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

      3. unpow2 34.7%

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

      4. unswap-sqr 46.7%

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

   8. Simplified 46.7%

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

      1. sqrt-div 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. sqr-pow 48.3%

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

       2. rem-sqrt-square 56.0%

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

       3. sqr-pow 55.9%

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

       4. fabs-sqr 55.9%

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

       5. sqr-pow 56.0%

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

       6. metadata-eval 56.0%

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

   12. Simplified 56.0%

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

   if 3.59999999999999994e87 < d

   1. Initial program 78.4%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in d around inf 68.9%

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

      1. associate-/r* 68.9%

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

   6. Simplified 68.9%

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

      1. sqrt-div 86.6%

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

   8. Applied egg-rr 86.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.05e-280)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* 0.5 (* M (/ D d))) 2.0)))))
   (if (<= d 6.2e-60)
     (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
     (if (<= d 2.2e+87)
       (*
        (sqrt (/ (* d (/ d h)) l))
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* (* 0.5 M) (/ D d)) 2.0)))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.05e-280) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 6.2e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else if (d <= 2.2e+87) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-2.05d-280)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((0.5d0 * (m * (d_1 / d))) ** 2.0d0))))
    else if (d <= 6.2d-60) then
        tmp = (-0.125d0) * ((((m * d_1) * (m * d_1)) * (sqrt(h) / (l ** 1.5d0))) / d)
    else if (d <= 2.2d+87) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * ((h / l) * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.05e-280) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((0.5 * (M * (D / d))), 2.0))));
	} else if (d <= 6.2e-60) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else if (d <= 2.2e+87) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * Math.pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.05e-280:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((0.5 * (M * (D / d))), 2.0))))
	elif d <= 6.2e-60:
		tmp = -0.125 * ((((M * D) * (M * D)) * (math.sqrt(h) / math.pow(l, 1.5))) / d)
	elif d <= 2.2e+87:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * math.pow(((0.5 * M) * (D / d)), 2.0))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.05e-280)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	elseif (d <= 6.2e-60)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	elseif (d <= 2.2e+87)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.05e-280)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((0.5 * (M * (D / d))) ^ 2.0))));
	elseif (d <= 6.2e-60)
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / (l ^ 1.5))) / d);
	elseif (d <= 2.2e+87)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * ((h / l) * (((0.5 * M) * (D / d)) ^ 2.0))));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.05e-280], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.2e-60], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e+87], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.0500000000000001e-280

   1. Initial program 74.3%

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

   3. Simplified 72.6%

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

      1. associate-*r/ 76.0%

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

      2. div-inv 76.0%

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

      3. metadata-eval 76.0%

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

   5. Applied egg-rr 76.0%

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

   7. Applied egg-rr 61.8%

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

      1. unpow1 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*r* 61.8%

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

   9. Simplified 61.8%

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

   if -2.0500000000000001e-280 < d < 6.19999999999999976e-60

   1. Initial program 43.6%

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

   3. Simplified 43.5%

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

      1. associate-*r/ 42.0%

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

      2. div-inv 42.0%

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

      3. metadata-eval 42.0%

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

   5. Applied egg-rr 42.0%

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

   6. Taylor expanded in d around 0 34.6%

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

      1. associate-*l/ 34.7%

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

      2. unpow2 34.7%

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

      3. unpow2 34.7%

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

      4. unswap-sqr 46.7%

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

   8. Simplified 46.7%

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

      1. sqrt-div 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. sqr-pow 48.3%

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

       2. rem-sqrt-square 56.0%

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

       3. sqr-pow 55.9%

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

       4. fabs-sqr 55.9%

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

       5. sqr-pow 56.0%

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

       6. metadata-eval 56.0%

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

   12. Simplified 56.0%

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

   if 6.19999999999999976e-60 < d < 2.2000000000000001e87

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Applied egg-rr 15.7%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 70.0%

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

      3. associate-*r/ 70.1%

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

      4. *-commutative 70.1%

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

   7. Simplified 70.1%

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

   if 2.2000000000000001e87 < d

   1. Initial program 78.4%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in d around inf 68.9%

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

      1. associate-/r* 68.9%

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

   6. Simplified 68.9%

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

      1. sqrt-div 86.6%

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

   8. Applied egg-rr 86.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11: 51.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -5.4e-161)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 4.5e-305)
       (* d (sqrt (cbrt (* t_0 (* t_0 t_0)))))
       (if (<= d 7.8e-56)
         (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -5.4e-161) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 4.5e-305) {
		tmp = d * sqrt(cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= 7.8e-56) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -5.4e-161) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 4.5e-305) {
		tmp = d * Math.sqrt(Math.cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= 7.8e-56) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -5.4e-161)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 4.5e-305)
		tmp = Float64(d * sqrt(cbrt(Float64(t_0 * Float64(t_0 * t_0)))));
	elseif (d <= 7.8e-56)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.4e-161], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-305], 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, 7.8e-56], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.3999999999999999e-161

   1. Initial program 76.9%

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

   3. Simplified 74.7%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.7%

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

   if -5.3999999999999999e-161 < d < 4.5000000000000002e-305

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in d around inf 24.4%

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

      1. add-cbrt-cube 45.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]

   6. Applied egg-rr 45.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]

   if 4.5000000000000002e-305 < d < 7.8e-56

   1. Initial program 48.6%

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

   3. Simplified 48.4%

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

      1. associate-*r/ 46.7%

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

      2. div-inv 46.7%

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

      3. metadata-eval 46.7%

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

   5. Applied egg-rr 46.7%

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

   6. Taylor expanded in d around 0 40.1%

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

      1. associate-*l/ 40.1%

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

      2. unpow2 40.1%

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

      3. unpow2 40.1%

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

      4. unswap-sqr 54.1%

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

   8. Simplified 54.1%

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

      1. sqrt-div 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. sqr-pow 56.1%

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

       2. rem-sqrt-square 64.9%

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

       3. sqr-pow 64.9%

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

       4. fabs-sqr 64.9%

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

       5. sqr-pow 64.9%

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

       6. metadata-eval 64.9%

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

   12. Simplified 64.9%

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

   if 7.8e-56 < d

   1. Initial program 79.1%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in d around inf 60.0%

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

      1. associate-/r* 61.1%

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

   6. Simplified 61.1%

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

      1. sqrt-div 75.9%

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

   8. Applied egg-rr 75.9%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-65}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -1e-162)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 9.5e-304)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (<= d 4.5e-65)
         (* -0.125 (/ (* (sqrt (/ h (pow l 3.0))) (* M (* M (* D D)))) d))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1e-162) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 9.5e-304) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if (d <= 4.5e-65) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * (M * (M * (D * D)))) / d);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1e-162) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 9.5e-304) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if (d <= 4.5e-65) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * (M * (M * (D * D)))) / d);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -1e-162)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 9.5e-304)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif (d <= 4.5e-65)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(M * Float64(M * Float64(D * D)))) / d));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e-162], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e-304], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-65], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.99999999999999954e-163

   1. Initial program 76.9%

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

   3. Simplified 74.7%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.7%

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

   if -9.99999999999999954e-163 < d < 9.50000000000000023e-304

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in d around inf 22.8%

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

      1. add-cbrt-cube 30.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 30.1%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Applied egg-rr 30.1%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

   if 9.50000000000000023e-304 < d < 4.4999999999999998e-65

   1. Initial program 50.6%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in d around 0 43.5%

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

      1. associate-*l/ 43.5%

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

      2. *-commutative 43.5%

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

      3. unpow2 43.5%

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

      4. associate-*l* 48.1%

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

      5. unpow2 48.1%

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

   6. Simplified 48.1%

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

   if 4.4999999999999998e-65 < d

   1. Initial program 79.4%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in d around inf 59.3%

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

      1. associate-/r* 60.4%

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

   6. Simplified 60.4%

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

      1. sqrt-div 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-65}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13: 48.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -1.25e-163)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 9.5e-304)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (<= d 2.1e-66)
         (* -0.125 (/ (* (* (* M D) (* M D)) (sqrt (/ h (pow l 3.0)))) d))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1.25e-163) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 9.5e-304) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if (d <= 2.1e-66) {
		tmp = -0.125 * ((((M * D) * (M * D)) * sqrt((h / pow(l, 3.0)))) / d);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1.25e-163) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 9.5e-304) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if (d <= 2.1e-66) {
		tmp = -0.125 * ((((M * D) * (M * D)) * Math.sqrt((h / Math.pow(l, 3.0)))) / d);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -1.25e-163)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 9.5e-304)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif (d <= 2.1e-66)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * sqrt(Float64(h / (l ^ 3.0)))) / d));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.25e-163], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e-304], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.1e-66], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.24999999999999994e-163

   1. Initial program 76.9%

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

   3. Simplified 74.7%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.7%

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

   if -1.24999999999999994e-163 < d < 9.50000000000000023e-304

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in d around inf 22.8%

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

      1. add-cbrt-cube 30.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 30.1%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Applied egg-rr 30.1%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

   if 9.50000000000000023e-304 < d < 2.1e-66

   1. Initial program 50.6%

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

   3. Simplified 50.4%

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

      1. associate-*r/ 48.5%

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

      2. div-inv 48.5%

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

      3. metadata-eval 48.5%

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

   5. Applied egg-rr 48.5%

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

   6. Taylor expanded in d around 0 43.5%

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

      1. associate-*l/ 43.5%

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

      2. unpow2 43.5%

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

      3. unpow2 43.5%

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

      4. unswap-sqr 58.6%

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

   8. Simplified 58.6%

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

   if 2.1e-66 < d

   1. Initial program 79.4%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in d around inf 59.3%

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

      1. associate-/r* 60.4%

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

   6. Simplified 60.4%

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

      1. sqrt-div 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14: 51.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -4.8e-161)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 4.5e-305)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (<= d 1.3e-57)
         (* -0.125 (/ (* (* (* M D) (* M D)) (/ (sqrt h) (pow l 1.5))) d))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -4.8e-161) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 4.5e-305) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if (d <= 1.3e-57) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (sqrt(h) / pow(l, 1.5))) / d);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -4.8e-161) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 4.5e-305) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if (d <= 1.3e-57) {
		tmp = -0.125 * ((((M * D) * (M * D)) * (Math.sqrt(h) / Math.pow(l, 1.5))) / d);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -4.8e-161)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 4.5e-305)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif (d <= 1.3e-57)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(sqrt(h) / (l ^ 1.5))) / d));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.8e-161], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-305], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-57], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.79999999999999998e-161

   1. Initial program 76.9%

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

   3. Simplified 74.7%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.7%

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

   if -4.79999999999999998e-161 < d < 4.5000000000000002e-305

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in d around inf 24.4%

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

      1. add-cbrt-cube 32.4%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 32.4%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Applied egg-rr 32.4%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

   if 4.5000000000000002e-305 < d < 1.29999999999999993e-57

   1. Initial program 48.6%

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

   3. Simplified 48.4%

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

      1. associate-*r/ 46.7%

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

      2. div-inv 46.7%

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

      3. metadata-eval 46.7%

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

   5. Applied egg-rr 46.7%

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

   6. Taylor expanded in d around 0 40.1%

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

      1. associate-*l/ 40.1%

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

      2. unpow2 40.1%

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

      3. unpow2 40.1%

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

      4. unswap-sqr 54.1%

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

   8. Simplified 54.1%

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

      1. sqrt-div 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. sqr-pow 56.1%

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

       2. rem-sqrt-square 64.9%

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

       3. sqr-pow 64.9%

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

       4. fabs-sqr 64.9%

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

       5. sqr-pow 64.9%

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

       6. metadata-eval 64.9%

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

   12. Simplified 64.9%

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

   if 1.29999999999999993e-57 < d

   1. Initial program 79.1%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in d around inf 60.0%

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

      1. associate-/r* 61.1%

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

   6. Simplified 61.1%

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

      1. sqrt-div 75.9%

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

   8. Applied egg-rr 75.9%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15: 44.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -1.85e-162)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 1.96e-230)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1.85e-162) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 1.96e-230) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -1.85e-162) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 1.96e-230) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -1.85e-162)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 1.96e-230)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e-162], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.96e-230], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.8500000000000001e-162

   1. Initial program 76.9%

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

   3. Simplified 74.7%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.7%

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

   if -1.8500000000000001e-162 < d < 1.96000000000000001e-230

   1. Initial program 47.0%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in d around inf 27.2%

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

      1. add-cbrt-cube 32.8%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 32.9%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

   if 1.96000000000000001e-230 < d

   1. Initial program 72.8%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in d around inf 47.0%

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

      1. associate-/r* 47.8%

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

   6. Simplified 47.8%

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

      1. sqrt-div 59.0%

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

   8. Applied egg-rr 59.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-282}:\\ \;\;\;\;d \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -8e-170)
     (* (- d) t_0)
     (if (<= d 4e-282) (* d t_0) (* d (* (pow h -0.5) (pow l -0.5)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -8e-170) {
		tmp = -d * t_0;
	} else if (d <= 4e-282) {
		tmp = d * t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-8d-170)) then
        tmp = -d * t_0
    else if (d <= 4d-282) then
        tmp = d * t_0
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -8e-170) {
		tmp = -d * t_0;
	} else if (d <= 4e-282) {
		tmp = d * t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -8e-170:
		tmp = -d * t_0
	elif d <= 4e-282:
		tmp = d * t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -8e-170)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= 4e-282)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -8e-170)
		tmp = -d * t_0;
	elseif (d <= 4e-282)
		tmp = d * t_0;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -8e-170], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, 4e-282], N[(d * t$95$0), $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 < -7.99999999999999987e-170

   1. Initial program 76.4%

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

   3. Simplified 74.0%

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

      1. add-cbrt-cube 55.0%

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

      2. pow3 55.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in d around -inf 54.2%

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

      1. associate-*r* 54.2%

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

      2. mul-1-neg 54.2%

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

      3. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   if -7.99999999999999987e-170 < d < 4.0000000000000001e-282

   1. Initial program 48.1%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in d around inf 23.5%

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

   if 4.0000000000000001e-282 < d

   1. Initial program 70.8%

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

   3. Simplified 70.9%

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

      1. associate-*r/ 72.6%

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

      2. div-inv 72.6%

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

      3. metadata-eval 72.6%

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

   5. Applied egg-rr 72.6%

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

   6. Taylor expanded in d around inf 46.8%

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

      1. unpow-1 46.8%

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

      2. sqr-pow 46.8%

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

      3. rem-sqrt-square 46.8%

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

      4. metadata-eval 46.8%

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

      5. sqr-pow 46.7%

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

      6. fabs-sqr 46.7%

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

      7. sqr-pow 46.8%

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

   8. Simplified 46.8%

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

      1. unpow-prod-down 57.9%

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

   10. Applied egg-rr 57.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-282}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-284}:\\ \;\;\;\;d \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -8e-170)
     (* (- d) t_0)
     (if (<= d 4e-284) (* d t_0) (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -8e-170) {
		tmp = -d * t_0;
	} else if (d <= 4e-284) {
		tmp = d * t_0;
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-8d-170)) then
        tmp = -d * t_0
    else if (d <= 4d-284) then
        tmp = d * t_0
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -8e-170) {
		tmp = -d * t_0;
	} else if (d <= 4e-284) {
		tmp = d * t_0;
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -8e-170:
		tmp = -d * t_0
	elif d <= 4e-284:
		tmp = d * t_0
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -8e-170)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= 4e-284)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -8e-170)
		tmp = -d * t_0;
	elseif (d <= 4e-284)
		tmp = d * t_0;
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -8e-170], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, 4e-284], N[(d * t$95$0), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.99999999999999987e-170

   1. Initial program 76.4%

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

   3. Simplified 74.0%

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

      1. add-cbrt-cube 55.0%

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

      2. pow3 55.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in d around -inf 54.2%

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

      1. associate-*r* 54.2%

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

      2. mul-1-neg 54.2%

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

      3. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   if -7.99999999999999987e-170 < d < 4.00000000000000015e-284

   1. Initial program 48.1%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in d around inf 23.5%

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

   if 4.00000000000000015e-284 < d

   1. Initial program 70.8%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in d around inf 46.8%

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

      1. associate-/r* 47.5%

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

   6. Simplified 47.5%

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

      1. sqrt-div 58.0%

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

   8. Applied egg-rr 58.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-284}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18: 44.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-282}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8e-170)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 5e-282)
     (* d (sqrt (/ 1.0 (* l h))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8e-170) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 5e-282) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-8d-170)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 5d-282) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8e-170) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 5e-282) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -8e-170:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 5e-282:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8e-170)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 5e-282)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8e-170)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 5e-282)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8e-170], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e-282], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.99999999999999987e-170

   1. Initial program 76.4%

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

   3. Simplified 74.1%

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

      1. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in M around 0 55.4%

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

   if -7.99999999999999987e-170 < d < 5.0000000000000001e-282

   1. Initial program 48.1%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in d around inf 23.5%

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

   if 5.0000000000000001e-282 < d

   1. Initial program 70.8%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in d around inf 46.8%

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

      1. associate-/r* 47.5%

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

   6. Simplified 47.5%

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

      1. sqrt-div 58.0%

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

   8. Applied egg-rr 58.0%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-282}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 19: 41.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.5e-170)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e-170) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-8.5d-170)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e-170) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.5e-170:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.5e-170)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.5e-170)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.5e-170], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.5e-170

   1. Initial program 76.4%

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

   3. Simplified 74.0%

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

      1. add-cbrt-cube 55.0%

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

      2. pow3 55.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in d around -inf 54.2%

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

      1. associate-*r* 54.2%

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

      2. mul-1-neg 54.2%

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

      3. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   if -8.5e-170 < d

   1. Initial program 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in d around inf 41.0%

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

      1. associate-/r* 41.6%

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

   6. Simplified 41.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 20: 26.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (l * h)))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Simplified 68.2%

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

4. Taylor expanded in d around inf 30.5%

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

5. Final simplification 30.5%

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

Alternative 21: 26.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / h) / l));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / h) / l))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Simplified 68.2%

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

4. Taylor expanded in d around inf 30.5%

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

   1. associate-/r* 30.9%

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

6. Simplified 30.9%

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

7. Final simplification 30.9%

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

Alternative 22: 26.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

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

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Simplified 68.2%

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

   1. associate-*r/ 70.5%

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

   2. div-inv 70.5%

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

   3. metadata-eval 70.5%

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

5. Applied egg-rr 70.5%

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

6. Taylor expanded in d around inf 30.5%

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

   1. unpow-1 30.5%

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

   2. sqr-pow 30.5%

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

   3. rem-sqrt-square 30.1%

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

   4. metadata-eval 30.1%

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

   5. sqr-pow 30.1%

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

   6. fabs-sqr 30.1%

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

   7. sqr-pow 30.1%

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

8. Simplified 30.1%

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

9. Final simplification 30.1%

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

Reproduce

herbie shell --seed 2023271 
(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)))))