Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.5% → 77.4%

Time: 22.7s

Alternatives: 15

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: 65.5% 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: 77.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 70.0%

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

   3. Simplified 69.2%

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

      1. frac-2neg 69.2%

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

      2. sqrt-div 79.6%

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

   5. Applied egg-rr 79.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 72.1%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 24.1%

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

   7. Simplified 80.8%

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

4. Final simplification 80.2%

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

Alternative 2: 74.7% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D 2.0) (/ M_m d)) 2.0))))
    (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l)))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (* M_m (/ 0.5 d))) 2.0)) 1.0)
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D / 2.0) * (M_m / d)), 2.0)))) * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m * (0.5 / d))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M_m / d)) ^ 2.0)))) * Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -5e-310], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 70.0%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 74.9%

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

   5. Applied egg-rr 74.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.1%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 24.1%

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

   7. Simplified 80.8%

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

4. Final simplification 78.1%

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

Alternative 3: 77.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 70.0%

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

   3. Simplified 70.9%

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

      1. frac-2neg 69.2%

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

      2. sqrt-div 79.6%

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

   5. Applied egg-rr 81.1%

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

   if -1.999999999999994e-310 < h

   1. Initial program 72.1%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 24.1%

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

   7. Simplified 80.8%

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

4. Final simplification 80.9%

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

Alternative 4: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{D}{\frac{d}{M_m}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \left(M_m \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 3.5e-305)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ D (/ d M_m)) 2.0))) l))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (* M_m (/ 0.5 d))) 2.0)) 1.0)
    (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 3.5e-305) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * (0.25 * pow((D / (d / M_m)), 2.0))) / l)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m * (0.5 / d))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 3.5e-305)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(D / Float64(d / M_m)) ^ 2.0))) / l))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 3.5e-305], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(D / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.4999999999999998e-305

   1. Initial program 70.0%

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

   3. Simplified 70.9%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-udef 70.5%

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

      3. *-commutative 70.5%

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

      4. frac-times 69.7%

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

      5. *-commutative 69.7%

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

      6. frac-times 68.9%

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

      7. div-inv 68.9%

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

      8. metadata-eval 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. expm1-def 68.9%

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

      2. expm1-log1p 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-*r* 69.2%

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

      6. metadata-eval 69.2%

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

      7. times-frac 69.2%

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

      8. associate-*r/ 69.2%

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

      9. associate-*r* 70.9%

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

      10. *-commutative 70.9%

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

      11. *-commutative 70.9%

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

      12. associate-/r* 70.9%

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

      13. metadata-eval 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in D around 0 70.0%

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

      1. add-sqr-sqrt 70.0%

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

      2. pow2 70.0%

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

      3. *-commutative 70.0%

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

      4. sqrt-prod 70.0%

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

      5. sqrt-pow1 70.0%

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

      6. metadata-eval 70.0%

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

      7. times-frac 70.0%

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

      8. *-un-lft-identity 70.0%

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

      9. frac-times 71.0%

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

      10. sqrt-pow1 70.9%

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

      11. sqrt-prod 70.9%

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

      12. pow2 70.9%

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

      13. add-sqr-sqrt 70.9%

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

      14. associate-*r/ 71.0%

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

   10. Applied egg-rr 71.0%

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

   if 3.4999999999999998e-305 < l

   1. Initial program 72.1%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 24.1%

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

   7. Simplified 80.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(D \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -5.7 \cdot 10^{-8}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + t_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* D (* 0.5 (/ M_m d))) 2.0)))
   (if (<= l -5.7e-8)
     (* (- d) (sqrt (/ (/ 1.0 l) h)))
     (if (<= l -5e-308)
       (* (sqrt (* (/ d l) (/ d h))) (+ 1.0 (* t_0 (* (/ h l) -0.5))))
       (* (/ (/ d (sqrt h)) (sqrt l)) (- 1.0 (* (/ h l) (* 0.5 t_0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((D * (0.5 * (M_m / d))), 2.0);
	double tmp;
	if (l <= -5.7e-8) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -5e-308) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (t_0 * ((h / l) * -0.5)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 * (0.5d0 * (m_m / d))) ** 2.0d0
    if (l <= (-5.7d-8)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= (-5d-308)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + (t_0 * ((h / l) * (-0.5d0))))
    else
        tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0d0 - ((h / l) * (0.5d0 * t_0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((D * (0.5 * (M_m / d))), 2.0);
	double tmp;
	if (l <= -5.7e-8) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -5e-308) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (t_0 * ((h / l) * -0.5)));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((D * (0.5 * (M_m / d))), 2.0)
	tmp = 0
	if l <= -5.7e-8:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= -5e-308:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (t_0 * ((h / l) * -0.5)))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * t_0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0
	tmp = 0.0
	if (l <= -5.7e-8)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -5e-308)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(t_0 * Float64(Float64(h / l) * -0.5))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (D * (0.5 * (M_m / d))) ^ 2.0;
	tmp = 0.0;
	if (l <= -5.7e-8)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= -5e-308)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (t_0 * ((h / l) * -0.5)));
	else
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -5.7e-8], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-308], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.70000000000000009e-8

   1. Initial program 61.4%

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

   3. Simplified 63.0%

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

      1. frac-2neg 59.8%

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

      2. sqrt-div 71.7%

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

   5. Applied egg-rr 74.5%

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

   6. Taylor expanded in d around -inf 57.1%

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

      1. associate-*r* 57.1%

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

      2. neg-mul-1 57.1%

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

      3. associate-/l/ 58.3%

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

   8. Simplified 58.3%

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

   if -5.70000000000000009e-8 < l < -4.99999999999999955e-308

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

      1. frac-2neg 79.1%

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

      2. sqrt-div 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. expm1-log1p-u 33.3%

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

      2. expm1-udef 30.5%

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

   7. Applied egg-rr 23.4%

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

      1. expm1-def 26.1%

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

      2. expm1-log1p 70.9%

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

      3. unpow1/2 70.9%

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

      4. associate-*r* 70.9%

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

      5. associate-*l* 70.9%

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

   9. Simplified 70.9%

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

   if -4.99999999999999955e-308 < l

   1. Initial program 72.1%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 24.1%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 84.8%

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

      3. associate-/r* 80.1%

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

      4. *-commutative 80.1%

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

      5. associate-*r/ 80.8%

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

      6. associate-*l* 80.8%

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

      7. *-commutative 80.8%

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

      8. associate-*l/ 77.3%

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

      9. *-commutative 77.3%

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

      10. associate-*l* 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{-8}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-167}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{D}{\frac{d}{M_m}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{\ell}}{h}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.6e-167)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ D (/ d M_m)) 2.0))) l))))
   (if (<= d 3.4e-292)
     (* d (sqrt (log1p (expm1 (/ (/ 1.0 l) h)))))
     (*
      (/ (/ d (sqrt h)) (sqrt l))
      (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M_m d))) 2.0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.6e-167) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * (0.25 * pow((D / (d / M_m)), 2.0))) / l)));
	} else if (d <= 3.4e-292) {
		tmp = d * sqrt(log1p(expm1(((1.0 / l) / h))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.6e-167) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * (0.25 * Math.pow((D / (d / M_m)), 2.0))) / l)));
	} else if (d <= 3.4e-292) {
		tmp = d * Math.sqrt(Math.log1p(Math.expm1(((1.0 / l) / h))));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -1.6e-167:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * (0.25 * math.pow((D / (d / M_m)), 2.0))) / l)))
	elif d <= 3.4e-292:
		tmp = d * math.sqrt(math.log1p(math.expm1(((1.0 / l) / h))))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M_m / d))), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.6e-167)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(D / Float64(d / M_m)) ^ 2.0))) / l))));
	elseif (d <= 3.4e-292)
		tmp = Float64(d * sqrt(log1p(expm1(Float64(Float64(1.0 / l) / h)))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.6e-167], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(D / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.4e-292], N[(d * N[Sqrt[N[Log[1 + N[(Exp[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e-167

   1. Initial program 77.1%

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

   3. Simplified 78.2%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 77.7%

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

      3. *-commutative 77.7%

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

      4. frac-times 76.7%

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

      5. *-commutative 76.7%

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

      6. frac-times 75.8%

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

      7. div-inv 75.8%

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

      8. metadata-eval 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. expm1-def 75.8%

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

      2. expm1-log1p 76.2%

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

      3. *-commutative 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. metadata-eval 76.2%

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

      7. times-frac 76.2%

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

      8. associate-*r/ 76.2%

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

      9. associate-*r* 78.2%

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

      10. *-commutative 78.2%

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

      11. *-commutative 78.2%

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

      12. associate-/r* 78.2%

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

      13. metadata-eval 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in D around 0 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. pow2 77.1%

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

      3. *-commutative 77.1%

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

      4. sqrt-prod 77.1%

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

      5. sqrt-pow1 77.1%

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

      6. metadata-eval 77.1%

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

      7. times-frac 77.1%

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

      8. *-un-lft-identity 77.1%

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

      9. frac-times 78.2%

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

      10. sqrt-pow1 78.2%

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

      11. sqrt-prod 78.2%

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

      12. pow2 78.2%

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

      13. add-sqr-sqrt 78.2%

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

      14. associate-*r/ 79.4%

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

   10. Applied egg-rr 79.4%

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

   if -1.6000000000000001e-167 < d < 3.40000000000000017e-292

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in d around inf 24.0%

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

      1. *-commutative 24.0%

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

      2. associate-/r* 24.0%

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

   6. Simplified 24.0%

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

      1. log1p-expm1-u 54.1%

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

   8. Applied egg-rr 54.1%

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

   if 3.40000000000000017e-292 < d

   1. Initial program 72.4%

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

   3. Simplified 68.8%

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

   5. Applied egg-rr 24.5%

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

      1. expm1-def 37.0%

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

      2. expm1-log1p 85.4%

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

      3. associate-/r* 80.5%

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

      4. *-commutative 80.5%

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

      5. associate-*r/ 81.2%

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

      6. associate-*l* 81.2%

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

      7. *-commutative 81.2%

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

      8. associate-*l/ 77.7%

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

      9. *-commutative 77.7%

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

      10. associate-*l* 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-167}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{\ell}}{h}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 7: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 3e-187)
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) -0.5))))
    (sqrt (/ d h)))
   (*
    (/ (/ d (sqrt h)) (sqrt l))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M_m d))) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 3e-187) {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5)))) * sqrt((d / h));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 3d-187) then
        tmp = (sqrt((d / l)) * (1.0d0 + ((h / l) * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))))) * sqrt((d / h))
    else
        tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m_m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 3e-187) {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h / l) * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5)))) * Math.sqrt((d / h));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 3e-187:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h / l) * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5)))) * math.sqrt((d / h))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M_m / d))), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 3e-187)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 3e-187)
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * ((((M_m / 2.0) * (D / d)) ^ 2.0) * -0.5)))) * sqrt((d / h));
	else
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M_m / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 3e-187], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.00000000000000004e-187

   1. Initial program 72.4%

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

   3. Simplified 71.7%

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

   if 3.00000000000000004e-187 < l

   1. Initial program 69.6%

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

   3. Simplified 65.5%

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

   5. Applied egg-rr 24.2%

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

      1. expm1-def 38.8%

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

      2. expm1-log1p 82.1%

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

      3. associate-/r* 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-*r/ 79.9%

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

      6. associate-*l* 79.9%

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

      7. *-commutative 79.9%

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

      8. associate-*l/ 75.8%

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

      9. *-commutative 75.8%

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

      10. associate-*l* 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-184}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 1.15e-184)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* D (* M_m (/ 0.5 d))) 2.0)))))
   (*
    (/ (/ d (sqrt h)) (sqrt l))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M_m d))) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.15e-184) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * pow((D * (M_m * (0.5 / d))), 2.0))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.15d-184) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h / l) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m_m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.15e-184) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * Math.pow((D * (M_m * (0.5 / d))), 2.0))));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M_m / d))), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 1.15e-184:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * math.pow((D * (M_m * (0.5 / d))), 2.0))))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M_m / d))), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 1.15e-184)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 1.15e-184)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * ((D * (M_m * (0.5 / d))) ^ 2.0))));
	else
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M_m / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 1.15e-184], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-184

   1. Initial program 72.4%

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

   3. Simplified 73.1%

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

      1. expm1-log1p-u 72.8%

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

      2. expm1-udef 72.8%

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

      3. *-commutative 72.8%

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

      4. frac-times 72.1%

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

      5. *-commutative 72.1%

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

      6. frac-times 71.5%

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

      7. div-inv 71.5%

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

      8. metadata-eval 71.5%

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

   5. Applied egg-rr 71.5%

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

      1. expm1-def 71.5%

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

      2. expm1-log1p 71.7%

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

      3. *-commutative 71.7%

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

      4. *-commutative 71.7%

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

      5. associate-*r* 71.7%

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

      6. metadata-eval 71.7%

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

      7. times-frac 71.7%

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

      8. associate-*r/ 71.7%

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

      9. associate-*r* 73.1%

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

      10. *-commutative 73.1%

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

      11. *-commutative 73.1%

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

      12. associate-/r* 73.1%

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

      13. metadata-eval 73.1%

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

   7. Simplified 73.1%

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

   if 1.15e-184 < l

   1. Initial program 69.6%

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

   3. Simplified 65.5%

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

   5. Applied egg-rr 24.2%

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

      1. expm1-def 38.8%

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

      2. expm1-log1p 82.1%

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

      3. associate-/r* 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-*r/ 79.9%

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

      6. associate-*l* 79.9%

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

      7. *-commutative 79.9%

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

      8. associate-*l/ 75.8%

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

      9. *-commutative 75.8%

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

      10. associate-*l* 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-184}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -3.8e-5)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 3.3e+177)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (+ 1.0 (* (pow (* D (* 0.5 (/ M_m d))) 2.0) (* (/ h l) -0.5))))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -3.8e-5) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 3.3e+177) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (pow((D * (0.5 * (M_m / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-3.8d-5)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 3.3d+177) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + (((d_1 * (0.5d0 * (m_m / d))) ** 2.0d0) * ((h / l) * (-0.5d0))))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -3.8e-5) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 3.3e+177) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (Math.pow((D * (0.5 * (M_m / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -3.8e-5:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 3.3e+177:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (math.pow((D * (0.5 * (M_m / d))), 2.0) * ((h / l) * -0.5)))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -3.8e-5)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 3.3e+177)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -3.8e-5)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 3.3e+177)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (((D * (0.5 * (M_m / d))) ^ 2.0) * ((h / l) * -0.5)));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -3.8e-5], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.3e+177], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8000000000000002e-5

   1. Initial program 61.4%

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

   3. Simplified 63.0%

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

      1. frac-2neg 59.8%

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

      2. sqrt-div 71.7%

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

   5. Applied egg-rr 74.5%

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

   6. Taylor expanded in d around -inf 57.1%

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

      1. associate-*r* 57.1%

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

      2. neg-mul-1 57.1%

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

      3. associate-/l/ 58.3%

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

   8. Simplified 58.3%

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

   if -3.8000000000000002e-5 < l < 3.3000000000000001e177

   1. Initial program 78.9%

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

   3. Simplified 76.0%

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

      1. frac-2neg 78.3%

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

      2. sqrt-div 30.0%

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

   5. Applied egg-rr 30.0%

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

      1. expm1-log1p-u 11.4%

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

      2. expm1-udef 10.4%

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

   7. Applied egg-rr 17.4%

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

      1. expm1-def 22.3%

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

      2. expm1-log1p 66.7%

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

      3. unpow1/2 66.7%

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

      4. associate-*r* 66.7%

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

      5. associate-*l* 66.7%

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

   9. Simplified 66.7%

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

   if 3.3000000000000001e177 < l

   1. Initial program 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in d around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-/r* 39.7%

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

   6. Simplified 39.7%

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

   7. Taylor expanded in d around 0 39.8%

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

      1. associate-/l/ 39.7%

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

      2. associate-/l/ 39.8%

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

      3. unpow-1 39.8%

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

      4. sqr-pow 39.8%

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

      5. rem-sqrt-square 39.8%

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

      6. metadata-eval 39.8%

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

      7. sqr-pow 39.6%

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

      8. fabs-sqr 39.6%

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

      9. sqr-pow 39.8%

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

   9. Simplified 39.8%

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

       1. add-sqr-sqrt 39.6%

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

       2. sqrt-unprod 39.8%

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

       3. pow-prod-up 39.8%

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

       4. metadata-eval 39.8%

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

       5. inv-pow 39.8%

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

       6. associate-/l/ 39.7%

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

       7. sqrt-div 62.5%

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

       8. inv-pow 62.5%

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

       9. sqrt-pow1 62.6%

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

       10. metadata-eval 62.6%

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

   11. Applied egg-rr 62.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 10: 43.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-279}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (- d) (sqrt (/ (/ 1.0 l) h)))))
   (if (<= l -8.5e-221)
     t_0
     (if (<= l 2.8e-279)
       (* d (pow (* h l) -0.5))
       (if (<= l 6e-138) t_0 (* d (/ (pow l -0.5) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = -d * sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -8.5e-221) {
		tmp = t_0;
	} else if (l <= 2.8e-279) {
		tmp = d * pow((h * l), -0.5);
	} else if (l <= 6e-138) {
		tmp = t_0;
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -d * sqrt(((1.0d0 / l) / h))
    if (l <= (-8.5d-221)) then
        tmp = t_0
    else if (l <= 2.8d-279) then
        tmp = d * ((h * l) ** (-0.5d0))
    else if (l <= 6d-138) then
        tmp = t_0
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = -d * Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -8.5e-221) {
		tmp = t_0;
	} else if (l <= 2.8e-279) {
		tmp = d * Math.pow((h * l), -0.5);
	} else if (l <= 6e-138) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = -d * math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= -8.5e-221:
		tmp = t_0
	elif l <= 2.8e-279:
		tmp = d * math.pow((h * l), -0.5)
	elif l <= 6e-138:
		tmp = t_0
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)))
	tmp = 0.0
	if (l <= -8.5e-221)
		tmp = t_0;
	elseif (l <= 2.8e-279)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	elseif (l <= 6e-138)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = -d * sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -8.5e-221)
		tmp = t_0;
	elseif (l <= 2.8e-279)
		tmp = d * ((h * l) ^ -0.5);
	elseif (l <= 6e-138)
		tmp = t_0;
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8.5e-221], t$95$0, If[LessEqual[l, 2.8e-279], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e-138], t$95$0, N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.49999999999999984e-221 or 2.8000000000000001e-279 < l < 6.0000000000000001e-138

   1. Initial program 72.3%

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

   3. Simplified 72.4%

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

      1. frac-2neg 71.6%

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

      2. sqrt-div 58.3%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in d around -inf 47.6%

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

      1. associate-*r* 47.6%

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

      2. neg-mul-1 47.6%

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

      3. associate-/l/ 48.1%

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

   8. Simplified 48.1%

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

   if -8.49999999999999984e-221 < l < 2.8000000000000001e-279

   1. Initial program 89.9%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in d around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/r* 36.7%

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

   6. Simplified 36.7%

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

   7. Taylor expanded in d around 0 36.7%

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

      1. associate-/l/ 36.7%

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

      2. associate-/l/ 36.7%

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

      3. unpow-1 36.7%

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

      4. sqr-pow 36.7%

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

      5. rem-sqrt-square 36.7%

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

      6. metadata-eval 36.7%

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

      7. sqr-pow 36.7%

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

      8. fabs-sqr 36.7%

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

      9. sqr-pow 36.7%

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

   9. Simplified 36.7%

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

   if 6.0000000000000001e-138 < l

   1. Initial program 65.9%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in d around inf 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 40.8%

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

   6. Simplified 40.8%

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

   7. Taylor expanded in d around 0 40.8%

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

      1. associate-/l/ 40.8%

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

      2. associate-/l/ 40.8%

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

      3. unpow-1 40.8%

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

      4. sqr-pow 40.9%

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

      5. rem-sqrt-square 40.9%

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

      6. metadata-eval 40.9%

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

      7. sqr-pow 40.7%

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

      8. fabs-sqr 40.7%

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

      9. sqr-pow 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 40.7%

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

       2. sqrt-unprod 40.9%

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

       3. pow-prod-up 40.8%

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

       4. metadata-eval 40.8%

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

       5. inv-pow 40.8%

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

       6. associate-/l/ 40.8%

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

       7. sqrt-div 48.1%

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

       8. inv-pow 48.1%

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

       9. sqrt-pow1 48.2%

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

       10. metadata-eval 48.2%

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

   11. Applied egg-rr 48.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-221}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-279}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 11: 43.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{\ell}}{h}\\ t_1 := \left(-d\right) \cdot \sqrt{t_0}\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-279}:\\ \;\;\;\;d \cdot \sqrt[3]{{t_0}^{1.5}}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 l) h)) (t_1 (* (- d) (sqrt t_0))))
   (if (<= l -7e-221)
     t_1
     (if (<= l 2.7e-279)
       (* d (cbrt (pow t_0 1.5)))
       (if (<= l 2.55e-138) t_1 (* d (/ (pow l -0.5) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (1.0 / l) / h;
	double t_1 = -d * sqrt(t_0);
	double tmp;
	if (l <= -7e-221) {
		tmp = t_1;
	} else if (l <= 2.7e-279) {
		tmp = d * cbrt(pow(t_0, 1.5));
	} else if (l <= 2.55e-138) {
		tmp = t_1;
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (1.0 / l) / h;
	double t_1 = -d * Math.sqrt(t_0);
	double tmp;
	if (l <= -7e-221) {
		tmp = t_1;
	} else if (l <= 2.7e-279) {
		tmp = d * Math.cbrt(Math.pow(t_0, 1.5));
	} else if (l <= 2.55e-138) {
		tmp = t_1;
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(1.0 / l) / h)
	t_1 = Float64(Float64(-d) * sqrt(t_0))
	tmp = 0.0
	if (l <= -7e-221)
		tmp = t_1;
	elseif (l <= 2.7e-279)
		tmp = Float64(d * cbrt((t_0 ^ 1.5)));
	elseif (l <= 2.55e-138)
		tmp = t_1;
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[((-d) * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7e-221], t$95$1, If[LessEqual[l, 2.7e-279], N[(d * N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.55e-138], t$95$1, N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.9999999999999998e-221 or 2.7000000000000001e-279 < l < 2.5500000000000001e-138

   1. Initial program 72.3%

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

   3. Simplified 72.4%

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

      1. frac-2neg 71.6%

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

      2. sqrt-div 58.3%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in d around -inf 47.6%

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

      1. associate-*r* 47.6%

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

      2. neg-mul-1 47.6%

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

      3. associate-/l/ 48.1%

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

   8. Simplified 48.1%

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

   if -6.9999999999999998e-221 < l < 2.7000000000000001e-279

   1. Initial program 89.9%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in d around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/r* 36.7%

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

   6. Simplified 36.7%

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

      1. add-cbrt-cube 46.4%

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

      2. pow1/3 46.4%

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

      3. add-sqr-sqrt 46.4%

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

      4. pow1 46.4%

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

      5. pow1/2 46.4%

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

      6. pow-prod-up 46.4%

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

      7. metadata-eval 46.4%

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

   8. Applied egg-rr 46.4%

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

      1. unpow1/3 46.4%

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

   10. Simplified 46.4%

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

   if 2.5500000000000001e-138 < l

   1. Initial program 65.9%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in d around inf 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 40.8%

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

   6. Simplified 40.8%

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

   7. Taylor expanded in d around 0 40.8%

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

      1. associate-/l/ 40.8%

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

      2. associate-/l/ 40.8%

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

      3. unpow-1 40.8%

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

      4. sqr-pow 40.9%

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

      5. rem-sqrt-square 40.9%

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

      6. metadata-eval 40.9%

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

      7. sqr-pow 40.7%

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

      8. fabs-sqr 40.7%

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

      9. sqr-pow 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 40.7%

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

       2. sqrt-unprod 40.9%

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

       3. pow-prod-up 40.8%

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

       4. metadata-eval 40.8%

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

       5. inv-pow 40.8%

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

       6. associate-/l/ 40.8%

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

       7. sqrt-div 48.1%

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

       8. inv-pow 48.1%

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

       9. sqrt-pow1 48.2%

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

       10. metadata-eval 48.2%

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

   11. Applied egg-rr 48.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-221}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-279}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 12: 40.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.26 \cdot 10^{-220} \lor \neg \left(\ell \leq 1.8 \cdot 10^{-279}\right) \land \ell \leq 6.6 \cdot 10^{-137}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (or (<= l -1.26e-220) (and (not (<= l 1.8e-279)) (<= l 6.6e-137)))
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (* d (pow (* h l) -0.5))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if ((l <= -1.26e-220) || (!(l <= 1.8e-279) && (l <= 6.6e-137))) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((l <= (-1.26d-220)) .or. (.not. (l <= 1.8d-279)) .and. (l <= 6.6d-137)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if ((l <= -1.26e-220) || (!(l <= 1.8e-279) && (l <= 6.6e-137))) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if (l <= -1.26e-220) or (not (l <= 1.8e-279) and (l <= 6.6e-137)):
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if ((l <= -1.26e-220) || (!(l <= 1.8e-279) && (l <= 6.6e-137)))
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if ((l <= -1.26e-220) || (~((l <= 1.8e-279)) && (l <= 6.6e-137)))
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[Or[LessEqual[l, -1.26e-220], And[N[Not[LessEqual[l, 1.8e-279]], $MachinePrecision], LessEqual[l, 6.6e-137]]], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.25999999999999992e-220 or 1.7999999999999998e-279 < l < 6.6000000000000004e-137

   1. Initial program 72.3%

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

   3. Simplified 72.4%

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

      1. frac-2neg 71.6%

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

      2. sqrt-div 58.3%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in d around -inf 47.6%

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

      1. associate-*r* 47.6%

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

      2. neg-mul-1 47.6%

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

      3. associate-/l/ 48.1%

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

   8. Simplified 48.1%

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

   if -1.25999999999999992e-220 < l < 1.7999999999999998e-279 or 6.6000000000000004e-137 < l

   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 66.6%

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

   4. Taylor expanded in d around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in d around 0 40.2%

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

      1. associate-/l/ 40.1%

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

      2. associate-/l/ 40.2%

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

      3. unpow-1 40.2%

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

      4. sqr-pow 40.2%

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

      5. rem-sqrt-square 40.2%

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

      6. metadata-eval 40.2%

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

      7. sqr-pow 40.0%

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

      8. fabs-sqr 40.0%

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

      9. sqr-pow 40.2%

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

   9. Simplified 40.2%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.26 \cdot 10^{-220} \lor \neg \left(\ell \leq 1.8 \cdot 10^{-279}\right) \land \ell \leq 6.6 \cdot 10^{-137}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 13: 40.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{-221} \lor \neg \left(\ell \leq 9 \cdot 10^{-280}\right) \land \ell \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (or (<= l -7e-221) (and (not (<= l 9e-280)) (<= l 2.6e-138)))
     (* d (- t_0))
     (* d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if ((l <= -7e-221) || (!(l <= 9e-280) && (l <= 2.6e-138))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if ((l <= (-7d-221)) .or. (.not. (l <= 9d-280)) .and. (l <= 2.6d-138)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if ((l <= -7e-221) || (!(l <= 9e-280) && (l <= 2.6e-138))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if (l <= -7e-221) or (not (l <= 9e-280) and (l <= 2.6e-138)):
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if ((l <= -7e-221) || (!(l <= 9e-280) && (l <= 2.6e-138)))
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if ((l <= -7e-221) || (~((l <= 9e-280)) && (l <= 2.6e-138)))
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[Or[LessEqual[l, -7e-221], And[N[Not[LessEqual[l, 9e-280]], $MachinePrecision], LessEqual[l, 2.6e-138]]], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.9999999999999998e-221 or 8.9999999999999991e-280 < l < 2.6e-138

   1. Initial program 72.3%

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

   3. Simplified 72.4%

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

      1. frac-2neg 71.6%

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

      2. sqrt-div 58.3%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in d around -inf 47.6%

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

      1. mul-1-neg 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-/l/ 48.1%

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

      4. distribute-rgt-neg-in 48.1%

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

      5. associate-/l/ 47.6%

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

      6. unpow-1 47.6%

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

      7. sqr-pow 47.6%

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

      8. rem-sqrt-square 47.6%

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

      9. metadata-eval 47.6%

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

      10. sqr-pow 47.4%

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

      11. fabs-sqr 47.4%

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

      12. sqr-pow 47.6%

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

   8. Simplified 47.6%

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

   if -6.9999999999999998e-221 < l < 8.9999999999999991e-280 or 2.6e-138 < l

   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 66.6%

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

   4. Taylor expanded in d around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in d around 0 40.2%

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

      1. associate-/l/ 40.1%

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

      2. associate-/l/ 40.2%

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

      3. unpow-1 40.2%

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

      4. sqr-pow 40.2%

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

      5. rem-sqrt-square 40.2%

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

      6. metadata-eval 40.2%

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

      7. sqr-pow 40.0%

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

      8. fabs-sqr 40.0%

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

      9. sqr-pow 40.2%

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

   9. Simplified 40.2%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-221} \lor \neg \left(\ell \leq 9 \cdot 10^{-280}\right) \land \ell \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 14: 26.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 69.6%

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

4. Taylor expanded in d around inf 24.0%

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

5. Final simplification 24.0%

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

Alternative 15: 26.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 69.6%

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

4. Taylor expanded in d around inf 24.0%

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

   1. *-commutative 24.0%

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

   2. associate-/r* 24.0%

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

6. Simplified 24.0%

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

7. Taylor expanded in d around 0 24.0%

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

   1. associate-/l/ 24.0%

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

   2. associate-/l/ 24.0%

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

   3. unpow-1 24.0%

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

   4. sqr-pow 24.0%

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

   5. rem-sqrt-square 23.7%

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

   6. metadata-eval 23.7%

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

   7. sqr-pow 23.6%

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

   8. fabs-sqr 23.6%

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

   9. sqr-pow 23.7%

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

9. Simplified 23.7%

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

10. Final simplification 23.7%

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

Reproduce

herbie shell --seed 2023338 
(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)))))