Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 81.4%

Time: 1.9min

Alternatives: 16

Speedup: 1.0×

• could not determine a ground truth

  1. d = -2.5906368375529228e-148
  2. h = 5.1720334873380296e+50
  3. l = -5.284431804405698e-23
  4. M = -1.190432651255918e-188
  5. D = 6.812114463054068e+296

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (pow (* (* D M_m) (/ 0.5 d)) 2.0) (/ l h)))))
   (*
    d
    (/
     (fma h (* (pow (* M_m (/ D (* d 2.0))) 2.0) (/ -0.5 l)) 1.0)
     (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * (pow(((D * M_m) * (0.5 / d)), 2.0) / (l / h))));
	} else {
		tmp = d * (fma(h, (pow((M_m * (D / (d * 2.0))), 2.0) * (-0.5 / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = Float64(d * Float64(fma(h, Float64((Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0) * Float64(-0.5 / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-309], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

      1. clear-num 69.1%

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

      2. un-div-inv 69.1%

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

      3. frac-times 70.5%

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

      4. associate-/l* 69.1%

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

      5. *-un-lft-identity 69.1%

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

      6. times-frac 69.1%

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

      7. metadata-eval 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. associate-*r* 69.1%

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

      2. associate-/l* 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-*r* 70.5%

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

      5. associate-/l* 70.5%

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

   8. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 80.3%

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

   10. Applied egg-rr 80.3%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. *-lft-identity 73.5%

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

      2. distribute-rgt-in 82.4%

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

      3. associate-*l/ 84.6%

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

      4. associate-/l* 84.5%

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

   8. Simplified 87.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2e-310)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* D (/ (/ M_m 2.0) d)) 2.0))))))
   (*
    d
    (/
     (fma h (* (pow (* M_m (/ D (* d 2.0))) 2.0) (/ -0.5 l)) 1.0)
     (* (sqrt l) (sqrt h))))))

C

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

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 80.3%

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

   6. Applied egg-rr 81.1%

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

   if -1.999999999999994e-310 < h

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. *-lft-identity 73.5%

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

      2. distribute-rgt-in 82.4%

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

      3. associate-*l/ 84.6%

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

      4. associate-/l* 84.5%

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

   8. Simplified 87.6%

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

4. Final simplification 84.5%

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

Alternative 3: 77.5% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2e-310)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
   (*
    d
    (/
     (fma h (* (pow (* M_m (/ D (* d 2.0))) 2.0) (/ -0.5 l)) 1.0)
     (* (sqrt l) (sqrt h))))))

C

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

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -2e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 71.4%

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

      2. frac-times 72.8%

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

      3. associate-/l* 71.4%

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

      4. *-un-lft-identity 71.4%

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

      5. times-frac 71.4%

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

      6. metadata-eval 71.4%

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

   6. Applied egg-rr 71.4%

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

   if -1.999999999999994e-310 < h

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. *-lft-identity 73.5%

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

      2. distribute-rgt-in 82.4%

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

      3. associate-*l/ 84.6%

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

      4. associate-/l* 84.5%

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

   8. Simplified 87.6%

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

4. Final simplification 80.0%

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

Alternative 4: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{0.5}{\ell} \cdot \left(h \cdot {\left(M\_m \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -2.2e-219)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (/ 0.5 l) (* h (pow (* M_m (* D (/ 0.5 d))) 2.0)))))
   (if (<= d 3.5e-308)
     (* d (sqrt (log1p (expm1 (/ 1.0 (* l h))))))
     (*
      (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
      (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -2.2e-219) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * pow((M_m * (D * (0.5 / d))), 2.0))));
	} else if (d <= 3.5e-308) {
		tmp = d * sqrt(log1p(expm1((1.0 / (l * h)))));
	} else {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -2.2e-219) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * Math.pow((M_m * (D * (0.5 / d))), 2.0))));
	} else if (d <= 3.5e-308) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1((1.0 / (l * h)))));
	} else {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -2.2e-219:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * math.pow((M_m * (D * (0.5 / d))), 2.0))))
	elif d <= 3.5e-308:
		tmp = d * math.sqrt(math.log1p(math.expm1((1.0 / (l * h)))))
	else:
		tmp = (1.0 + (math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -2.2e-219)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 / l) * Float64(h * (Float64(M_m * Float64(D * Float64(0.5 / d))) ^ 2.0)))));
	elseif (d <= 3.5e-308)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(1.0 / Float64(l * h))))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -2.2e-219], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.5 / l), $MachinePrecision] * N[(h * N[Power[N[(M$95$m * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-308], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.1999999999999999e-219

   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 74.7%

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

      1. clear-num 74.7%

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

      2. un-div-inv 74.7%

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

      3. frac-times 76.4%

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

      4. associate-/l* 74.7%

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

      5. *-un-lft-identity 74.7%

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

      6. times-frac 74.7%

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

      7. metadata-eval 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. associate-*r* 74.7%

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

      2. associate-/l* 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-*r* 76.4%

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

      5. associate-/l* 76.4%

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

   8. Simplified 76.4%

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

      1. clear-num 76.4%

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

      2. inv-pow 76.4%

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

   10. Applied egg-rr 76.4%

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

       1. unpow-1 76.4%

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

       2. associate-/l/ 79.2%

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

       3. associate-*l* 79.2%

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

   12. Simplified 79.2%

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

       1. pow1 79.2%

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

       2. sqrt-unprod 70.0%

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

       3. un-div-inv 70.0%

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

       4. *-commutative 70.0%

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

       5. associate-*r* 70.0%

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

       6. *-commutative 70.0%

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

   14. Applied egg-rr 70.0%

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

       1. unpow1 70.0%

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

       2. associate-/r/ 70.0%

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

       3. associate-*r* 69.0%

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

   16. Simplified 69.0%

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

   if -2.1999999999999999e-219 < d < 3.5e-308

   1. Initial program 40.3%

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

   3. Simplified 40.3%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 2.5%

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

      1. associate-/l/ 2.5%

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

   9. Simplified 2.5%

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

       1. associate-/l/ 2.5%

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

       2. *-commutative 2.5%

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

       3. inv-pow 2.5%

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

       4. metadata-eval 2.5%

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

       5. pow-prod-up 2.5%

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

       6. log1p-expm1-u 42.6%

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

       7. pow-prod-up 42.6%

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

       8. metadata-eval 42.6%

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

       9. inv-pow 42.6%

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

       10. *-commutative 42.6%

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

   11. Applied egg-rr 42.6%

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

   if 3.5e-308 < d

   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 69.4%

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

   6. Applied egg-rr 74.1%

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

      1. distribute-rgt1-in 83.0%

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

      2. +-commutative 83.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r* 83.0%

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

      5. associate-/l* 84.5%

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

      6. *-commutative 84.5%

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

      7. associate-*r* 84.5%

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

      8. associate-/l* 84.5%

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

      9. associate-/r* 84.5%

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

   8. Simplified 84.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{0.5}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell \cdot h}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 1e-307)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* D (/ (/ M_m 2.0) d)) 2.0))))
     (sqrt (/ d h))))
   (*
    (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1e-307) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * pow((D * ((M_m / 2.0) / d)), 2.0)))) * sqrt((d / h)));
	} else {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1d-307) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * ((-0.5d0) * ((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0)))) * sqrt((d / h)))
    else
        tmp = (1.0d0 + ((((d_1 * m_m) * (0.5d0 / d)) ** 2.0d0) * ((-0.5d0) * (h / l)))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1e-307) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * Math.pow((D * ((M_m / 2.0) / d)), 2.0)))) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 1e-307:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * math.pow((D * ((M_m / 2.0) / d)), 2.0)))) * math.sqrt((d / h)))
	else:
		tmp = (1.0 + (math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 1e-307)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0)))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 1e-307)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * ((D * ((M_m / 2.0) / d)) ^ 2.0)))) * sqrt((d / h)));
	else
		tmp = (1.0 + ((((D * M_m) * (0.5 / d)) ^ 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 1e-307], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999909e-308

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

   if 9.99999999999999909e-308 < l

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. distribute-rgt1-in 82.4%

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

      2. +-commutative 82.4%

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

      3. associate-*r* 82.4%

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

      4. associate-*r* 82.4%

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

      5. associate-/l* 83.9%

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

      6. *-commutative 83.9%

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

      7. associate-*r* 83.9%

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

      8. associate-/l* 83.9%

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

      9. associate-/r* 83.9%

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

   8. Simplified 83.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
   (*
    (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-309)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = (1.0d0 + ((((d_1 * m_m) * (0.5d0 / d)) ** 2.0d0) * ((-0.5d0) * (h / l)))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1e-309:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	else:
		tmp = (1.0 + (math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	else
		tmp = (1.0 + ((((D * M_m) * (0.5 / d)) ^ 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. distribute-rgt1-in 82.4%

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

      2. +-commutative 82.4%

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

      3. associate-*r* 82.4%

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

      4. associate-*r* 82.4%

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

      5. associate-/l* 83.9%

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

      6. *-commutative 83.9%

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

      7. associate-*r* 83.9%

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

      8. associate-/l* 83.9%

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

      9. associate-/r* 83.9%

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

   8. Simplified 83.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 2e-309)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
   (*
    (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2e-309) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2d-309) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = (1.0d0 + ((((d_1 * m_m) * (0.5d0 / d)) ** 2.0d0) * ((-0.5d0) * (h / l)))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2e-309) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 2e-309:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = (1.0 + (math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 2e-309)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 2e-309)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = (1.0 + ((((D * M_m) * (0.5 / d)) ^ 2.0) * (-0.5 * (h / l)))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 2e-309], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9999999999999988e-309

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 71.4%

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

      2. frac-times 72.8%

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

      3. associate-/l* 71.4%

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

      4. *-un-lft-identity 71.4%

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

      5. times-frac 71.4%

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

      6. metadata-eval 71.4%

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

   6. Applied egg-rr 71.4%

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

   if 1.9999999999999988e-309 < l

   1. Initial program 70.3%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 73.5%

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

      1. distribute-rgt1-in 82.4%

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

      2. +-commutative 82.4%

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

      3. associate-*r* 82.4%

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

      4. associate-*r* 82.4%

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

      5. associate-/l* 83.9%

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

      6. *-commutative 83.9%

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

      7. associate-*r* 83.9%

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

      8. associate-/l* 83.9%

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

      9. associate-/r* 83.9%

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

   8. Simplified 83.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-102}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+189}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.7e-102)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -1e-309)
     (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
     (if (<= l 1.3e+189)
       (*
        (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
        (/ d (sqrt (* l h))))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.7e-102) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -1e-309) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else if (l <= 1.3e+189) {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * (d / sqrt((l * h)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.7e-102) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -1e-309) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else if (l <= 1.3e+189) {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.7e-102)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -1e-309)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	elseif (l <= 1.3e+189)
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.7e-102], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -1e-309], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+189], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.70000000000000006e-102

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 48.2%

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

      6. neg-mul-1 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in l around 0 48.2%

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

      1. associate-/l/ 48.8%

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

   10. Simplified 48.8%

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

   if -1.70000000000000006e-102 < l < -1.000000000000002e-309

   1. Initial program 75.0%

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

   3. Simplified 71.4%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 20.8%

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

      1. associate-/l/ 20.8%

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

   9. Simplified 20.8%

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

       1. add-cbrt-cube 31.2%

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

       2. add-sqr-sqrt 31.2%

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

       3. pow1 31.2%

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

       4. pow1/2 31.2%

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

       5. pow-prod-up 31.2%

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

       6. associate-/l/ 31.2%

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

       7. metadata-eval 31.2%

          \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}} \]

   11. Applied egg-rr 31.2%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

   if -1.000000000000002e-309 < l < 1.29999999999999991e189

   1. Initial program 77.2%

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

   3. Simplified 74.4%

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

   6. Applied egg-rr 74.8%

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

      1. distribute-rgt1-in 86.5%

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

      2. +-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. associate-*r* 86.5%

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

      5. associate-/l* 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-*r* 89.4%

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

      8. associate-/l* 89.4%

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

      9. associate-/r* 89.4%

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

   8. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. associate-/l/ 89.4%

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

      3. pow1/2 89.4%

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

      4. pow1/2 89.4%

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

      5. pow-prod-down 82.0%

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

      6. *-commutative 82.0%

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

      7. pow1/2 82.0%

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

      8. *-commutative 82.0%

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

   10. Applied egg-rr 82.0%

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

       1. *-lft-identity 82.0%

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

   12. Simplified 82.0%

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

   if 1.29999999999999991e189 < l

   1. Initial program 49.2%

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

   3. Simplified 52.0%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in d around inf 53.1%

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

      1. associate-/l/ 53.2%

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

   9. Simplified 53.2%

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

       1. sqrt-div 67.6%

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

   11. Applied egg-rr 67.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-102}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+189}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{0.5}{\ell} \cdot \left(h \cdot {\left(M\_m \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+189}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (/ 0.5 l) (* h (pow (* M_m (* D (/ 0.5 d))) 2.0)))))
   (if (<= l 1e+189)
     (*
      (+ 1.0 (* (pow (* (* D M_m) (/ 0.5 d)) 2.0) (* -0.5 (/ h l))))
      (/ d (sqrt (* l h))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * pow((M_m * (D * (0.5 / d))), 2.0))));
	} else if (l <= 1e+189) {
		tmp = (1.0 + (pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * (d / sqrt((l * h)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-309)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((0.5d0 / l) * (h * ((m_m * (d_1 * (0.5d0 / d))) ** 2.0d0))))
    else if (l <= 1d+189) then
        tmp = (1.0d0 + ((((d_1 * m_m) * (0.5d0 / d)) ** 2.0d0) * ((-0.5d0) * (h / l)))) * (d / sqrt((l * h)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-309) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * Math.pow((M_m * (D * (0.5 / d))), 2.0))));
	} else if (l <= 1e+189) {
		tmp = (1.0 + (Math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1e-309:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * math.pow((M_m * (D * (0.5 / d))), 2.0))))
	elif l <= 1e+189:
		tmp = (1.0 + (math.pow(((D * M_m) * (0.5 / d)), 2.0) * (-0.5 * (h / l)))) * (d / math.sqrt((l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 / l) * Float64(h * (Float64(M_m * Float64(D * Float64(0.5 / d))) ^ 2.0)))));
	elseif (l <= 1e+189)
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((0.5 / l) * (h * ((M_m * (D * (0.5 / d))) ^ 2.0))));
	elseif (l <= 1e+189)
		tmp = (1.0 + ((((D * M_m) * (0.5 / d)) ^ 2.0) * (-0.5 * (h / l)))) * (d / sqrt((l * h)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-309], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.5 / l), $MachinePrecision] * N[(h * N[Power[N[(M$95$m * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+189], N[(N[(1.0 + N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 69.1%

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

      1. clear-num 69.1%

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

      2. un-div-inv 69.1%

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

      3. frac-times 70.5%

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

      4. associate-/l* 69.1%

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

      5. *-un-lft-identity 69.1%

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

      6. times-frac 69.1%

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

      7. metadata-eval 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. associate-*r* 69.1%

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

      2. associate-/l* 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-*r* 70.5%

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

      5. associate-/l* 70.5%

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

   8. Simplified 70.5%

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

      1. clear-num 70.4%

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

      2. inv-pow 70.4%

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

   10. Applied egg-rr 70.4%

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

       1. unpow-1 70.4%

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

       2. associate-/l/ 72.8%

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

       3. associate-*l* 72.8%

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

   12. Simplified 72.8%

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

       1. pow1 72.8%

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

       2. sqrt-unprod 61.2%

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

       3. un-div-inv 61.2%

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

       4. *-commutative 61.2%

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

       5. associate-*r* 61.2%

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

       6. *-commutative 61.2%

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

   14. Applied egg-rr 61.2%

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

       1. unpow1 61.2%

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

       2. associate-/r/ 61.2%

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

       3. associate-*r* 60.4%

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

   16. Simplified 60.4%

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

   if -1.000000000000002e-309 < l < 1e189

   1. Initial program 77.2%

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

   3. Simplified 74.4%

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

   6. Applied egg-rr 74.8%

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

      1. distribute-rgt1-in 86.5%

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

      2. +-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. associate-*r* 86.5%

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

      5. associate-/l* 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-*r* 89.4%

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

      8. associate-/l* 89.4%

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

      9. associate-/r* 89.4%

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

   8. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. associate-/l/ 89.4%

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

      3. pow1/2 89.4%

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

      4. pow1/2 89.4%

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

      5. pow-prod-down 82.0%

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

      6. *-commutative 82.0%

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

      7. pow1/2 82.0%

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

      8. *-commutative 82.0%

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

   10. Applied egg-rr 82.0%

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

       1. *-lft-identity 82.0%

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

   12. Simplified 82.0%

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

   if 1e189 < l

   1. Initial program 49.2%

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

   3. Simplified 52.0%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in d around inf 53.1%

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

      1. associate-/l/ 53.2%

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

   9. Simplified 53.2%

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

       1. sqrt-div 67.6%

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

   11. Applied egg-rr 67.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{0.5}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+189}:\\ \;\;\;\;\left(1 + {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{if}\;\ell \leq -2.52 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-250}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-155}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (cbrt (pow (/ 1.0 (* l h)) 1.5))))
   (if (<= l -2.52e-104)
     (* d (- (sqrt (/ (/ 1.0 l) h))))
     (if (<= l -6e-250)
       (* d t_0)
       (if (<= l 2.3e-155)
         (* (- d) t_0)
         (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = cbrt(pow((1.0 / (l * h)), 1.5));
	double tmp;
	if (l <= -2.52e-104) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -6e-250) {
		tmp = d * t_0;
	} else if (l <= 2.3e-155) {
		tmp = -d * t_0;
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	double tmp;
	if (l <= -2.52e-104) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -6e-250) {
		tmp = d * t_0;
	} else if (l <= 2.3e-155) {
		tmp = -d * t_0;
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5))
	tmp = 0.0
	if (l <= -2.52e-104)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -6e-250)
		tmp = Float64(d * t_0);
	elseif (l <= 2.3e-155)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[l, -2.52e-104], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -6e-250], N[(d * t$95$0), $MachinePrecision], If[LessEqual[l, 2.3e-155], N[((-d) * t$95$0), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.52e-104

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 48.2%

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

      6. neg-mul-1 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in l around 0 48.2%

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

      1. associate-/l/ 48.8%

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

   10. Simplified 48.8%

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

   if -2.52e-104 < l < -6.00000000000000032e-250

   1. Initial program 75.5%

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

   3. Simplified 73.6%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 21.1%

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

      1. associate-/l/ 21.1%

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

   9. Simplified 21.1%

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

       1. add-cbrt-cube 31.9%

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

       2. add-sqr-sqrt 31.9%

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

       3. pow1 31.9%

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

       4. pow1/2 31.9%

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

       5. pow-prod-up 31.9%

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

       6. associate-/l/ 31.9%

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

       7. metadata-eval 31.9%

          \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}} \]

   11. Applied egg-rr 31.9%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

   if -6.00000000000000032e-250 < l < 2.30000000000000005e-155

   1. Initial program 76.6%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 29.8%

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

      6. neg-mul-1 29.8%

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

   7. Simplified 29.8%

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

      1. add-cbrt-cube 37.4%

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

      2. pow1/3 37.4%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333}} \cdot \left(-d\right) \]

      3. add-sqr-sqrt 37.4%

         \[\leadsto {\left(\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333} \cdot \left(-d\right) \]

      4. pow1 37.4%

         \[\leadsto {\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333} \cdot \left(-d\right) \]

      5. pow1/2 37.4%

         \[\leadsto {\left({\left(\frac{1}{\ell \cdot h}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)}^{0.3333333333333333} \cdot \left(-d\right) \]

      6. pow-prod-up 37.4%

         \[\leadsto {\color{blue}{\left({\left(\frac{1}{\ell \cdot h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \cdot \left(-d\right) \]

      7. associate-/r* 37.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \cdot \left(-d\right) \]

      8. metadata-eval 37.4%

         \[\leadsto {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(-d\right) \]

   9. Applied egg-rr 37.4%

      \[\leadsto \color{blue}{{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-d\right) \]
   10. Step-by-step derivation

       1. unpow1/3 37.4%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}}} \cdot \left(-d\right) \]

       2. associate-/l/ 37.4%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{1.5}} \cdot \left(-d\right) \]

   11. Simplified 37.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \cdot \left(-d\right) \]

   if 2.30000000000000005e-155 < l

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   6. Applied egg-rr 52.9%

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

   7. Taylor expanded in d around inf 46.8%

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

      1. associate-/l/ 46.9%

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

   9. Simplified 46.9%

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

       1. sqrt-div 54.7%

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

   11. Applied egg-rr 54.7%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.52 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-155}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-268}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.02e-103)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5.3e-250)
     (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
     (if (<= l 5.5e-268)
       (* d (- (pow (* l h) -0.5)))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.02e-103) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5.3e-250) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else if (l <= 5.5e-268) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.02e-103) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -5.3e-250) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else if (l <= 5.5e-268) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.02e-103)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5.3e-250)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	elseif (l <= 5.5e-268)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.02e-103], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5.3e-250], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e-268], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.01999999999999998e-103

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 48.2%

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

      6. neg-mul-1 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in l around 0 48.2%

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

      1. associate-/l/ 48.8%

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

   10. Simplified 48.8%

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

   if -1.01999999999999998e-103 < l < -5.3000000000000001e-250

   1. Initial program 75.5%

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

   3. Simplified 73.6%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 21.1%

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

      1. associate-/l/ 21.1%

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

   9. Simplified 21.1%

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

       1. add-cbrt-cube 31.9%

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

       2. add-sqr-sqrt 31.9%

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

       3. pow1 31.9%

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

       4. pow1/2 31.9%

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

       5. pow-prod-up 31.9%

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

       6. associate-/l/ 31.9%

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

       7. metadata-eval 31.9%

          \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}} \]

   11. Applied egg-rr 31.9%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

   if -5.3000000000000001e-250 < l < 5.4999999999999997e-268

   1. Initial program 78.5%

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

   3. Simplified 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. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 40.4%

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

      6. neg-mul-1 40.4%

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

   7. Simplified 40.4%

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

      1. pow1/2 40.4%

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

      2. inv-pow 40.4%

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

      3. pow-pow 40.4%

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

      4. metadata-eval 40.4%

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

   9. Applied egg-rr 40.4%

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

   if 5.4999999999999997e-268 < l

   1. Initial program 69.1%

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

   3. Simplified 67.5%

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

   6. Applied egg-rr 45.3%

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

   7. Taylor expanded in d around inf 40.1%

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

      1. associate-/l/ 40.1%

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

   9. Simplified 40.1%

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

       1. sqrt-div 47.7%

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

   11. Applied egg-rr 47.7%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-268}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 10^{-302}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h 1e-302)
   (* d (- (pow (* l h) -0.5)))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= 1e-302) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 1d-302) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= 1e-302) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= 1e-302:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= 1e-302)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= 1e-302)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, 1e-302], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.9999999999999996e-303

   1. Initial program 70.7%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 38.5%

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

      6. neg-mul-1 38.5%

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

   7. Simplified 38.5%

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

      1. pow1/2 38.5%

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

      2. inv-pow 38.5%

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

      3. pow-pow 39.2%

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

      4. metadata-eval 39.2%

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

   9. Applied egg-rr 39.2%

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

   if 9.9999999999999996e-303 < h

   1. Initial program 70.1%

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

   3. Simplified 68.7%

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

   6. Applied egg-rr 42.4%

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

   7. Taylor expanded in d around inf 37.6%

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

      1. associate-/l/ 37.6%

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

   9. Simplified 37.6%

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

       1. sqrt-div 44.5%

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

   11. Applied egg-rr 44.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 10^{-302}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq 1.65 \cdot 10^{-260}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= d 1.65e-260) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (d <= 1.65e-260) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (d <= 1.65d-260) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (d <= 1.65e-260) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if d <= 1.65e-260:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= 1.65e-260)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (d <= 1.65e-260)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, 1.65e-260], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.6499999999999999e-260

   1. Initial program 66.0%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 36.8%

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

      6. neg-mul-1 36.8%

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

   7. Simplified 36.8%

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

      1. pow1/2 36.8%

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

      2. inv-pow 36.8%

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

      3. pow-pow 37.3%

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

      4. metadata-eval 37.3%

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

   9. Applied egg-rr 37.3%

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

   if 1.6499999999999999e-260 < d

   1. Initial program 75.5%

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

   3. Simplified 73.8%

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

      1. clear-num 73.9%

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

      2. un-div-inv 73.9%

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

      3. frac-times 75.5%

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

      4. associate-/l* 73.9%

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

      5. *-un-lft-identity 73.9%

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

      6. times-frac 73.9%

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

      7. metadata-eval 73.9%

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

   6. Applied egg-rr 73.9%

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

      1. associate-*r* 73.9%

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

      2. associate-/l* 75.5%

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

      3. *-commutative 75.5%

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

      4. associate-*r* 75.5%

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

      5. associate-/l* 75.5%

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

   8. Simplified 75.5%

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

      1. clear-num 75.5%

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

      2. inv-pow 75.5%

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

   10. Applied egg-rr 75.5%

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

       1. unpow-1 75.5%

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

       2. associate-/l/ 74.0%

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

       3. associate-*l* 73.2%

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

   12. Simplified 73.2%

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

   13. Taylor expanded in d around inf 40.4%

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

       1. unpow-1 40.4%

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

       2. metadata-eval 40.4%

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

       3. pow-sqr 40.4%

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

       4. rem-sqrt-square 41.1%

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

       5. metadata-eval 41.1%

          \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

       6. pow-sqr 41.0%

          \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \]

       7. fabs-sqr 41.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \]

       8. pow-sqr 41.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \]

       9. metadata-eval 41.1%

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

   15. Simplified 41.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{-260}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt((1.0 / (l * h)))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

3. Simplified 69.0%

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

5. Taylor expanded in d around inf 25.0%

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

   1. *-commutative 25.0%

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

7. Simplified 25.0%

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

8. Final simplification 25.0%

   \[\leadsto d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
9. Add Preprocessing

Alternative 15: 26.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

3. Simplified 69.0%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in d around inf 25.0%

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

   1. associate-/l/ 25.0%

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

9. Simplified 25.0%

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

10. Final simplification 25.0%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
11. Add Preprocessing

Alternative 16: 25.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (pow (* l h) -0.5)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.pow((l * h), -0.5)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

3. Simplified 69.0%

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

   1. clear-num 69.0%

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

   2. un-div-inv 69.0%

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

   3. frac-times 70.4%

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

   4. associate-/l* 69.0%

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

   5. *-un-lft-identity 69.0%

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

   6. times-frac 69.0%

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

   7. metadata-eval 69.0%

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

6. Applied egg-rr 69.0%

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

   1. associate-*r* 69.0%

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

   2. associate-/l* 70.4%

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

   3. *-commutative 70.4%

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

   4. associate-*r* 70.4%

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

   5. associate-/l* 70.4%

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

8. Simplified 70.4%

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

   1. clear-num 70.4%

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

   2. inv-pow 70.4%

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

10. Applied egg-rr 70.4%

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

    1. unpow-1 70.4%

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

    2. associate-/l/ 70.8%

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

    3. associate-*l* 70.4%

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

12. Simplified 70.4%

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

13. Taylor expanded in d around inf 25.0%

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

    1. unpow-1 25.0%

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

    2. metadata-eval 25.0%

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

    3. pow-sqr 25.0%

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

    4. rem-sqrt-square 24.6%

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

    5. metadata-eval 24.6%

       \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

    6. pow-sqr 24.5%

       \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \]

    7. fabs-sqr 24.5%

       \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \]

    8. pow-sqr 24.6%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \]

    9. metadata-eval 24.6%

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

15. Simplified 24.6%

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

16. Final simplification 24.6%

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

Reproduce

herbie shell --seed 2024053 
(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)))))