Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 81.4%

Time: 23.1s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{t\_0}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, \left(t\_0 \cdot 0.25\right) \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* D_m (/ M_m d)) 2.0)))
   (if (<= l -4e-310)
     (*
      (sqrt (/ d h))
      (* (/ (sqrt (- d)) (sqrt (- l))) (- 1.0 (* h (* 0.125 (/ t_0 l))))))
     (* d (/ (fma h (* (* t_0 0.25) (/ -0.5 l)) 1.0) (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((D_m * (M_m / d)), 2.0);
	double tmp;
	if (l <= -4e-310) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 - (h * (0.125 * (t_0 / l)))));
	} else {
		tmp = d * (fma(h, ((t_0 * 0.25) * (-0.5 / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d)) ^ 2.0
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 / l))))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(Float64(t_0 * 0.25) * Float64(-0.5 / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[(t$95$0 * 0.25), $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 < -3.999999999999988e-310

   1. Initial program 75.4%

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

   3. Simplified 75.0%

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

      1. associate-*r/ 75.4%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in h around -inf 49.1%

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

      1. associate-*r* 49.1%

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

      2. neg-mul-1 49.1%

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

      3. sub-neg 49.1%

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

      4. distribute-lft-in 49.1%

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

   9. Simplified 79.5%

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

       1. frac-2neg 79.5%

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

       2. sqrt-div 85.9%

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

   11. Applied egg-rr 85.9%

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

   if -3.999999999999988e-310 < l

   1. Initial program 64.3%

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

   3. Simplified 63.5%

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

   6. Applied egg-rr 78.6%

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

   8. Simplified 86.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.25\right) \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot 0.25\right) \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 3e-309)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* h (* 0.125 (/ (pow (/ (* D_m M_m) d) 2.0) l))))))
   (*
    d
    (/
     (fma h (* (* (pow (* D_m (/ M_m d)) 2.0) 0.25) (/ -0.5 l)) 1.0)
     (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3e-309) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (pow(((D_m * M_m) / d), 2.0) / l)))));
	} else {
		tmp = d * (fma(h, ((pow((D_m * (M_m / d)), 2.0) * 0.25) * (-0.5 / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 3e-309)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) / l))))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * 0.25) * Float64(-0.5 / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 3e-309], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $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 < 3.000000000000001e-309

   1. Initial program 75.4%

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

   3. Simplified 75.0%

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

      1. associate-*r/ 75.4%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in h around -inf 49.1%

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

      1. associate-*r* 49.1%

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

      2. neg-mul-1 49.1%

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

      3. sub-neg 49.1%

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

      4. distribute-lft-in 49.1%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in D around 0 80.2%

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

   if 3.000000000000001e-309 < l

   1. Initial program 64.3%

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

   3. Simplified 63.5%

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

   6. Applied egg-rr 78.6%

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

   8. Simplified 86.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.25\right) \cdot \frac{-0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{-41}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-187}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))))
   (if (<= d -1.1e-41)
     (* t_0 t_1)
     (if (<= d 1.55e-187)
       (* t_0 (* t_1 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ -0.125 l)))))
       (* d (/ (pow h -0.5) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.1e-41) {
		tmp = t_0 * t_1;
	} else if (d <= 1.55e-187) {
		tmp = t_0 * (t_1 * (h * (pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    if (d <= (-1.1d-41)) then
        tmp = t_0 * t_1
    else if (d <= 1.55d-187) then
        tmp = t_0 * (t_1 * (h * (((d_m * (m_m / d)) ** 2.0d0) * ((-0.125d0) / l))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.1e-41) {
		tmp = t_0 * t_1;
	} else if (d <= 1.55e-187) {
		tmp = t_0 * (t_1 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.1e-41:
		tmp = t_0 * t_1
	elif d <= 1.55e-187:
		tmp = t_0 * (t_1 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.1e-41)
		tmp = Float64(t_0 * t_1);
	elseif (d <= 1.55e-187)
		tmp = Float64(t_0 * Float64(t_1 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(-0.125 / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.1e-41)
		tmp = t_0 * t_1;
	elseif (d <= 1.55e-187)
		tmp = t_0 * (t_1 * (h * (((D_m * (M_m / d)) ^ 2.0) * (-0.125 / l))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.1e-41], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[d, 1.55e-187], N[(t$95$0 * N[(t$95$1 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1e-41

   1. Initial program 78.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in M around 0 63.1%

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

   if -1.1e-41 < d < 1.5500000000000001e-187

   1. Initial program 58.3%

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

   3. Simplified 58.2%

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

      1. associate-*r/ 58.3%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in M around inf 25.8%

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

      1. associate-*r/ 25.8%

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

      2. associate-*r* 29.3%

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

      3. associate-*r* 29.3%

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

      4. associate-*l/ 29.4%

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

      5. associate-*r/ 29.4%

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

      6. *-commutative 29.4%

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

      7. associate-*r/ 29.4%

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

      8. *-commutative 29.4%

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

      9. times-frac 29.4%

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

      10. *-commutative 29.4%

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

      11. associate-/l* 29.4%

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

      12. unpow2 29.4%

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

      13. unpow2 29.4%

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

      14. unpow2 29.4%

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

      15. times-frac 38.3%

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

      16. swap-sqr 46.1%

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

      17. unpow2 46.1%

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

   9. Simplified 43.9%

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

   if 1.5500000000000001e-187 < d

   1. Initial program 71.7%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in d around inf 59.3%

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

      1. associate-/r* 60.3%

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

   7. Simplified 60.3%

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

      1. sqrt-div 67.0%

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

      2. inv-pow 67.0%

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

      3. sqrt-pow1 67.0%

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

      4. metadata-eval 67.0%

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

   9. Applied egg-rr 67.0%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -9 \cdot 10^{-47}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-189}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(-0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))))
   (if (<= d -9e-47)
     (* t_0 t_1)
     (if (<= d 1.2e-189)
       (* t_0 (* t_1 (* -0.125 (* (pow (* D_m (/ M_m d)) 2.0) (/ h l)))))
       (* d (/ (pow h -0.5) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -9e-47) {
		tmp = t_0 * t_1;
	} else if (d <= 1.2e-189) {
		tmp = t_0 * (t_1 * (-0.125 * (pow((D_m * (M_m / d)), 2.0) * (h / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    if (d <= (-9d-47)) then
        tmp = t_0 * t_1
    else if (d <= 1.2d-189) then
        tmp = t_0 * (t_1 * ((-0.125d0) * (((d_m * (m_m / d)) ** 2.0d0) * (h / l))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -9e-47) {
		tmp = t_0 * t_1;
	} else if (d <= 1.2e-189) {
		tmp = t_0 * (t_1 * (-0.125 * (Math.pow((D_m * (M_m / d)), 2.0) * (h / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -9e-47:
		tmp = t_0 * t_1
	elif d <= 1.2e-189:
		tmp = t_0 * (t_1 * (-0.125 * (math.pow((D_m * (M_m / d)), 2.0) * (h / l))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -9e-47)
		tmp = Float64(t_0 * t_1);
	elseif (d <= 1.2e-189)
		tmp = Float64(t_0 * Float64(t_1 * Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -9e-47)
		tmp = t_0 * t_1;
	elseif (d <= 1.2e-189)
		tmp = t_0 * (t_1 * (-0.125 * (((D_m * (M_m / d)) ^ 2.0) * (h / l))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -9e-47], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[d, 1.2e-189], N[(t$95$0 * N[(t$95$1 * N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9e-47

   1. Initial program 77.7%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in M around 0 62.3%

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

   if -9e-47 < d < 1.1999999999999999e-189

   1. Initial program 59.1%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in M around inf 25.9%

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

      1. associate-*r* 29.7%

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

      2. times-frac 29.7%

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

      3. *-commutative 29.7%

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

      4. associate-/l* 29.7%

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

      5. unpow2 29.7%

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

      6. unpow2 29.7%

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

      7. unpow2 29.7%

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

      8. times-frac 37.5%

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

      9. swap-sqr 46.6%

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

      10. unpow2 46.6%

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

      11. associate-*r/ 46.7%

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

      12. *-commutative 46.7%

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

      13. associate-/l* 43.1%

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

   7. Simplified 43.1%

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

   if 1.1999999999999999e-189 < d

   1. Initial program 71.7%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in d around inf 59.3%

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

      1. associate-/r* 60.3%

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

   7. Simplified 60.3%

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

      1. sqrt-div 67.0%

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

      2. inv-pow 67.0%

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

      3. sqrt-pow1 67.0%

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

      4. metadata-eval 67.0%

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

   9. Applied egg-rr 67.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\left(D\_m \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.9e-290)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* h (* 0.125 (/ (pow (/ (* D_m M_m) d) 2.0) l))))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* (* -0.5 (/ h l)) (pow (* (* D_m M_m) (/ 0.5 d)) 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.9e-290) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (pow(((D_m * M_m) / d), 2.0) / l)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * pow(((D_m * M_m) * (0.5 / d)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 1.9d-290) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (h * (0.125d0 * ((((d_m * m_m) / d) ** 2.0d0) / l)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((-0.5d0) * (h / l)) * (((d_m * m_m) * (0.5d0 / d)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.9e-290) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * (0.125 * (Math.pow(((D_m * M_m) / d), 2.0) / l)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * Math.pow(((D_m * M_m) * (0.5 / d)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1.9e-290:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * (0.125 * (math.pow(((D_m * M_m) / d), 2.0) / l)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * math.pow(((D_m * M_m) * (0.5 / d)), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.9e-290)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) / l))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * (Float64(Float64(D_m * M_m) * Float64(0.5 / d)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1.9e-290)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * ((((D_m * M_m) / d) ^ 2.0) / l)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * (((D_m * M_m) * (0.5 / d)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.9e-290], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.89999999999999988e-290

   1. Initial program 75.2%

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

   3. Simplified 74.8%

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

      1. associate-*r/ 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in h around -inf 49.5%

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

      1. associate-*r* 49.5%

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

      2. neg-mul-1 49.5%

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

      3. sub-neg 49.5%

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

      4. distribute-lft-in 49.5%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in D around 0 80.7%

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

   if 1.89999999999999988e-290 < l

   1. Initial program 64.2%

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

   3. Simplified 63.4%

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

      1. add-sqr-sqrt 63.4%

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

      2. pow2 63.4%

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

   6. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   8. Applied egg-rr 78.9%

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

   10. Simplified 82.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-287}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\left(D\_m \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1e-287)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ (pow (* D_m (/ M_m d)) 2.0) l) (* h -0.125)))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* (* -0.5 (/ h l)) (pow (* (* D_m M_m) (/ 0.5 d)) 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1e-287) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * pow(((D_m * M_m) * (0.5 / d)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 1d-287) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((((d_m * (m_m / d)) ** 2.0d0) / l) * (h * (-0.125d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((-0.5d0) * (h / l)) * (((d_m * m_m) * (0.5d0 / d)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1e-287) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((Math.pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * Math.pow(((D_m * M_m) * (0.5 / d)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1e-287:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((math.pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * math.pow(((D_m * M_m) * (0.5 / d)), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1e-287)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l) * Float64(h * -0.125)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * (Float64(Float64(D_m * M_m) * Float64(0.5 / d)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1e-287)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((((D_m * (M_m / d)) ^ 2.0) / l) * (h * -0.125))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((-0.5 * (h / l)) * (((D_m * M_m) * (0.5 / d)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1e-287], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000002e-287

   1. Initial program 75.2%

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

   3. Simplified 74.8%

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

      1. associate-*r/ 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in h around -inf 49.5%

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

      1. associate-*r* 49.5%

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

      2. neg-mul-1 49.5%

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

      3. sub-neg 49.5%

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

      4. distribute-lft-in 49.5%

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

   9. Simplified 80.0%

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

       1. expm1-log1p-u 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)} + 1\right)\right) \]

       2. expm1-undefine 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)} - 1\right)} + 1\right)\right) \]

       3. associate-*r* 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}\right)} - 1\right) + 1\right)\right) \]

   11. Applied egg-rr 50.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} - 1\right)} + 1\right)\right) \]
   12. Step-by-step derivation

       1. sub-neg 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} + \left(-1\right)\right)} + 1\right)\right) \]

       2. metadata-eval 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} + \color{blue}{-1}\right) + 1\right)\right) \]

       3. +-commutative 50.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)}\right)} + 1\right)\right) \]

       4. log1p-undefine 50.2%

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

       5. rem-exp-log 80.0%

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

       6. associate-+r+ 80.0%

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

       7. metadata-eval 80.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{0} + \left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]

       8. +-lft-identity 80.0%

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

       9. *-commutative 80.0%

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

       10. neg-mul-1 80.0%

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

       11. associate-*r* 80.0%

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

       12. metadata-eval 80.0%

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

   13. Simplified 80.0%

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

   if 1.00000000000000002e-287 < l

   1. Initial program 64.2%

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

   3. Simplified 63.4%

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

      1. add-sqr-sqrt 63.4%

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

      2. pow2 63.4%

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

   6. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   8. Applied egg-rr 78.9%

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

   10. Simplified 82.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-287}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 9.5e+150)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ (pow (* D_m (/ M_m d)) 2.0) l) (* h -0.125)))))
   (* d (* (pow h -0.5) (sqrt (/ 1.0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e+150) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))));
	} else {
		tmp = d * (pow(h, -0.5) * sqrt((1.0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (d <= 9.5d+150) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((((d_m * (m_m / d)) ** 2.0d0) / l) * (h * (-0.125d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e+150) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((Math.pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 9.5e+150:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((math.pow((D_m * (M_m / d)), 2.0) / l) * (h * -0.125))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.sqrt((1.0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 9.5e+150)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l) * Float64(h * -0.125)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 9.5e+150)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((((D_m * (M_m / d)) ^ 2.0) / l) * (h * -0.125))));
	else
		tmp = d * ((h ^ -0.5) * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 9.5e+150], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.5000000000000001e150

   1. Initial program 71.4%

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

   3. Simplified 71.1%

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

      1. associate-*r/ 71.4%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in h around -inf 46.4%

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

      1. associate-*r* 46.4%

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

      2. neg-mul-1 46.4%

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

      3. sub-neg 46.4%

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

      4. distribute-lft-in 46.4%

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

   9. Simplified 73.9%

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

       1. expm1-log1p-u 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)} + 1\right)\right) \]

       2. expm1-undefine 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)} - 1\right)} + 1\right)\right) \]

       3. associate-*r* 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}\right)} - 1\right) + 1\right)\right) \]

   11. Applied egg-rr 45.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} - 1\right)} + 1\right)\right) \]
   12. Step-by-step derivation

       1. sub-neg 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} + \left(-1\right)\right)} + 1\right)\right) \]

       2. metadata-eval 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} + \color{blue}{-1}\right) + 1\right)\right) \]

       3. +-commutative 45.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)}\right)} + 1\right)\right) \]

       4. log1p-undefine 45.0%

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

       5. rem-exp-log 73.9%

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

       6. associate-+r+ 73.9%

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

       7. metadata-eval 73.9%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{0} + \left(\left(-h\right) \cdot 0.125\right) \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]

       8. +-lft-identity 73.9%

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

       9. *-commutative 73.9%

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

       10. neg-mul-1 73.9%

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

       11. associate-*r* 73.9%

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

       12. metadata-eval 73.9%

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

   13. Simplified 73.9%

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

   if 9.5000000000000001e150 < d

   1. Initial program 52.6%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in d around inf 90.6%

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

      1. associate-/r* 90.8%

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

   7. Simplified 90.8%

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

      1. div-inv 90.7%

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

      2. sqrt-prod 95.1%

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

      3. inv-pow 95.1%

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

      4. sqrt-pow1 95.2%

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

      5. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(D\_m \cdot M\_m\right)}^{2} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{-d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.9e-55)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -5e-310)
     (* -0.125 (/ (* (pow (* D_m M_m) 2.0) (sqrt (/ h (pow l 3.0)))) (- d)))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-55) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = -0.125 * ((pow((D_m * M_m), 2.0) * sqrt((h / pow(l, 3.0)))) / -d);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-2.9d-55)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-5d-310)) then
        tmp = (-0.125d0) * ((((d_m * m_m) ** 2.0d0) * sqrt((h / (l ** 3.0d0)))) / -d)
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.9e-55) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = -0.125 * ((Math.pow((D_m * M_m), 2.0) * Math.sqrt((h / Math.pow(l, 3.0)))) / -d);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.9e-55:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -5e-310:
		tmp = -0.125 * ((math.pow((D_m * M_m), 2.0) * math.sqrt((h / math.pow(l, 3.0)))) / -d)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.9e-55)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = Float64(-0.125 * Float64(Float64((Float64(D_m * M_m) ^ 2.0) * sqrt(Float64(h / (l ^ 3.0)))) / Float64(-d)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.9e-55)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -5e-310)
		tmp = -0.125 * ((((D_m * M_m) ^ 2.0) * sqrt((h / (l ^ 3.0)))) / -d);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.9e-55], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(-0.125 * N[(N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.9e-55

   1. Initial program 78.2%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in M around 0 62.0%

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

   if -2.9e-55 < d < -4.999999999999985e-310

   1. Initial program 69.9%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 37.8%

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

      6. associate-/l* 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l/ 0.0%

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

      2. associate-*r* 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. swap-sqr 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 47.1%

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

      9. associate-*l* 47.1%

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

      10. mul-1-neg 47.1%

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

   10. Simplified 47.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in d around inf 48.4%

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

      1. associate-/r* 49.1%

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

   7. Simplified 49.1%

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

      1. sqrt-div 54.9%

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

      2. inv-pow 54.9%

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

      3. sqrt-pow1 55.0%

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

      4. metadata-eval 55.0%

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

   9. Applied egg-rr 55.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{-d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({\left(D\_m \cdot M\_m\right)}^{2} \cdot \frac{0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.25e-54)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -5e-310)
     (* (sqrt (/ h (pow l 3.0))) (* (pow (* D_m M_m) 2.0) (/ 0.125 d)))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.25e-54) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (pow((D_m * M_m), 2.0) * (0.125 / d));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-1.25d-54)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-5d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * (((d_m * m_m) ** 2.0d0) * (0.125d0 / d))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.25e-54) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (Math.pow((D_m * M_m), 2.0) * (0.125 / d));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.25e-54:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -5e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (math.pow((D_m * M_m), 2.0) * (0.125 / d))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.25e-54)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64((Float64(D_m * M_m) ^ 2.0) * Float64(0.125 / d)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.25e-54)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -5e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (((D_m * M_m) ^ 2.0) * (0.125 / d));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.25e-54], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.25000000000000004e-54

   1. Initial program 78.2%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in M around 0 62.0%

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

   if -1.25000000000000004e-54 < d < -4.999999999999985e-310

   1. Initial program 69.9%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 37.8%

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

      6. associate-/l* 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. unpow2 0.0%

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

      5. unpow2 0.0%

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

      6. swap-sqr 0.0%

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

      7. unpow2 0.0%

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

      8. unpow2 0.0%

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

      9. rem-square-sqrt 46.9%

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

      10. *-commutative 46.9%

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

      11. associate-*l/ 46.9%

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

      12. associate-*r* 46.9%

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

      13. associate-*r/ 46.9%

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

      14. metadata-eval 46.9%

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

   10. Simplified 46.9%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in d around inf 48.4%

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

      1. associate-/r* 49.1%

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

   7. Simplified 49.1%

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

      1. sqrt-div 54.9%

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

      2. inv-pow 54.9%

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

      3. sqrt-pow1 55.0%

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

      4. metadata-eval 55.0%

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

   9. Applied egg-rr 55.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{\ell \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -3.2e-186)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -5e-310)
     (/ d (log (exp (sqrt (* l h)))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -3.2e-186) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = d / log(exp(sqrt((l * h))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-3.2d-186)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-5d-310)) then
        tmp = d / log(exp(sqrt((l * h))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -3.2e-186) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = d / Math.log(Math.exp(Math.sqrt((l * h))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -3.2e-186:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -5e-310:
		tmp = d / math.log(math.exp(math.sqrt((l * h))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -3.2e-186)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = Float64(d / log(exp(sqrt(Float64(l * h)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -3.2e-186)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -5e-310)
		tmp = d / log(exp(sqrt((l * h))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -3.2e-186], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d / N[Log[N[Exp[N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.2e-186

   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 75.0%

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

   5. Taylor expanded in M around 0 55.9%

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

   if -3.2e-186 < d < -4.999999999999985e-310

   1. Initial program 74.9%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in d around inf 26.8%

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

      1. associate-/r* 26.8%

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

   7. Simplified 26.8%

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

      1. pow1 26.8%

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

      2. associate-/l/ 26.8%

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

      3. sqrt-div 23.3%

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

      4. metadata-eval 23.3%

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

   9. Applied egg-rr 23.3%

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

       1. unpow1 23.3%

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

       2. associate-*r/ 23.3%

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

       3. *-rgt-identity 23.3%

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

       4. *-commutative 23.3%

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

   11. Simplified 23.3%

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

       1. add-log-exp 54.5%

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

   13. Applied egg-rr 54.5%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in d around inf 48.4%

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

      1. associate-/r* 49.1%

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

   7. Simplified 49.1%

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

      1. sqrt-div 54.9%

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

      2. inv-pow 54.9%

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

      3. sqrt-pow1 55.0%

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

      4. metadata-eval 55.0%

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

   9. Applied egg-rr 55.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{\ell \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.85e+168)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l 2.85e-230)
     (* d (- (pow (* l h) -0.5)))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.85e+168) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= 2.85e-230) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-2.85d+168)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= 2.85d-230) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.85e+168) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= 2.85e-230) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.85e+168:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= 2.85e-230:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.85e+168)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= 2.85e-230)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.85e+168)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= 2.85e-230)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.85e+168], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.85e-230], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.85000000000000019e168

   1. Initial program 64.0%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in M around 0 60.6%

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

   if -2.85000000000000019e168 < l < 2.84999999999999984e-230

   1. Initial program 77.7%

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

   3. Simplified 75.4%

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

      1. add-sqr-sqrt 75.4%

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

      2. pow2 75.4%

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

   6. Applied egg-rr 75.9%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-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. unpow1/2 0.0%

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

      4. rem-exp-log 0.0%

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

      5. exp-neg 0.0%

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

      6. exp-prod 0.0%

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

      7. distribute-lft-neg-out 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. exp-to-pow 0.0%

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

      11. *-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 47.2%

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

      15. neg-mul-1 47.2%

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

   9. Simplified 47.2%

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

   if 2.84999999999999984e-230 < l

   1. Initial program 64.0%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in d around inf 51.6%

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

      1. associate-/r* 52.4%

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

   7. Simplified 52.4%

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

      1. sqrt-div 58.8%

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

      2. inv-pow 58.8%

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

      3. sqrt-pow1 58.8%

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

      4. metadata-eval 58.8%

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

   9. Applied egg-rr 58.8%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2.8e-230)
   (* d (- (pow (* l h) -0.5)))
   (* d (/ (pow h -0.5) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.8e-230) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 2.8d-230) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.8e-230) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 2.8e-230:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2.8e-230)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 2.8e-230)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2.8e-230], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8000000000000001e-230

   1. Initial program 74.7%

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

   3. Simplified 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
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.6%

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

      2. pow2 73.6%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-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. unpow1/2 0.0%

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

      4. rem-exp-log 0.0%

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

      5. exp-neg 0.0%

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

      6. exp-prod 0.0%

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

      7. distribute-lft-neg-out 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. exp-to-pow 0.0%

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

      11. *-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 45.1%

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

      15. neg-mul-1 45.1%

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

   9. Simplified 45.1%

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

   if 2.8000000000000001e-230 < l

   1. Initial program 64.0%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in d around inf 51.6%

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

      1. associate-/r* 52.4%

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

   7. Simplified 52.4%

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

      1. sqrt-div 58.8%

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

      2. inv-pow 58.8%

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

      3. sqrt-pow1 58.8%

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

      4. metadata-eval 58.8%

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

   9. Applied egg-rr 58.8%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2.8e-230) (* d (- (pow (* l h) -0.5))) (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.8e-230) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 2.8d-230) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.8e-230) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 2.8e-230:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2.8e-230)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 2.8e-230)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2.8e-230], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8000000000000001e-230

   1. Initial program 74.7%

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

   3. Simplified 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
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.6%

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

      2. pow2 73.6%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-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. unpow1/2 0.0%

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

      4. rem-exp-log 0.0%

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

      5. exp-neg 0.0%

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

      6. exp-prod 0.0%

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

      7. distribute-lft-neg-out 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. exp-to-pow 0.0%

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

      11. *-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 45.1%

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

      15. neg-mul-1 45.1%

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

   9. Simplified 45.1%

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

   if 2.8000000000000001e-230 < l

   1. Initial program 64.0%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in d around inf 51.6%

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

      1. associate-/r* 52.4%

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

   7. Simplified 52.4%

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

      1. pow1 52.4%

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

      2. associate-/l/ 51.6%

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

      3. sqrt-div 51.6%

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

      4. metadata-eval 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. unpow1 51.6%

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

       2. associate-*r/ 51.6%

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

       3. *-rgt-identity 51.6%

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

       4. *-commutative 51.6%

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

   11. Simplified 51.6%

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

       1. sqrt-prod 58.8%

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

   13. Applied egg-rr 58.8%

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

       1. *-commutative 58.8%

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

   15. Simplified 58.8%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-230}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.15 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.15e-238)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.15e-238) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-5.15d-238)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.15e-238) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.15e-238:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.15e-238)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.15e-238)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.15e-238], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.1500000000000002e-238

   1. Initial program 75.9%

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

   3. Simplified 75.4%

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

      1. add-sqr-sqrt 75.4%

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

      2. pow2 75.4%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-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. unpow1/2 0.0%

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

      4. rem-exp-log 0.0%

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

      5. exp-neg 0.0%

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

      6. exp-prod 0.0%

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

      7. distribute-lft-neg-out 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. exp-to-pow 0.0%

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

      11. *-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 51.2%

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

      15. neg-mul-1 51.2%

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

   9. Simplified 51.2%

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

   if -5.1500000000000002e-238 < d

   1. Initial program 65.0%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in d around inf 46.6%

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

      1. associate-/r* 47.3%

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

   7. Simplified 47.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.15 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Simplified 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. Taylor expanded in d around inf 29.9%

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

   1. associate-/r* 30.3%

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

7. Simplified 30.3%

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

Alternative 16: 26.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Simplified 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. Taylor expanded in d around inf 29.9%

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

6. Final simplification 29.9%

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

Alternative 17: 26.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Simplified 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. add-sqr-sqrt 69.1%

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

   2. pow2 69.1%

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

6. Applied egg-rr 69.4%

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

7. Taylor expanded in d around inf 29.9%

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

   1. *-commutative 29.9%

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

   2. unpow1/2 29.9%

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

   3. rem-exp-log 28.8%

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

   4. exp-neg 28.8%

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

   5. exp-prod 28.4%

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

   6. distribute-lft-neg-out 28.4%

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

   7. distribute-rgt-neg-in 28.4%

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

   8. metadata-eval 28.4%

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

   9. exp-to-pow 29.6%

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

   10. *-commutative 29.6%

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

9. Simplified 29.6%

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

10. Final simplification 29.6%

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

Alternative 18: 26.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Simplified 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. Taylor expanded in d around inf 29.9%

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

   1. associate-/r* 30.3%

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

7. Simplified 30.3%

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

   1. pow1 30.3%

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

   2. associate-/l/ 29.9%

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

   3. sqrt-div 29.6%

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

   4. metadata-eval 29.6%

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

9. Applied egg-rr 29.6%

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

    1. unpow1 29.6%

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

    2. associate-*r/ 29.6%

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

    3. *-rgt-identity 29.6%

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

    4. *-commutative 29.6%

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

11. Simplified 29.6%

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

12. Final simplification 29.6%

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

Reproduce

herbie shell --seed 2024094 
(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)))))