Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.0% → 79.7%

Time: 1.6min

Alternatives: 18

Speedup: 1.0×

• could not determine a ground truth

  1. d = -1.4270902577512459e-207
  2. h = 2.692879766906599e+247
  3. l = -1.2558999538483664e+21
  4. M = 2.0431480845514602e+134
  5. D = -8.022611797214432e-227

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 0.6× speedup

Math

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

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 (<= h -1e-310)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (/ D (* d (/ 2.0 M_m))) 2.0) -0.5)))))
   (*
    (/ (sqrt d) (sqrt h))
    (*
     (* (sqrt d) (sqrt (/ 1.0 l)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M_m 2.0) (/ D 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 (h <= -1e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (pow((D / (d * (2.0 / M_m))), 2.0) * -0.5))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) * sqrt((1.0 / l))) * (1.0 + ((h / l) * (-0.5 * pow(((M_m / 2.0) * (D / 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 (h <= (-1d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 + ((h / l) * (((d_1 / (d * (2.0d0 / m_m))) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) * sqrt((1.0d0 / l))) * (1.0d0 + ((h / l) * ((-0.5d0) * (((m_m / 2.0d0) * (d_1 / 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 (h <= -1e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 + ((h / l) * (Math.pow((D / (d * (2.0 / M_m))), 2.0) * -0.5))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) * Math.sqrt((1.0 / l))) * (1.0 + ((h / l) * (-0.5 * Math.pow(((M_m / 2.0) * (D / 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 h <= -1e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 + ((h / l) * (math.pow((D / (d * (2.0 / M_m))), 2.0) * -0.5))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) * math.sqrt((1.0 / l))) * (1.0 + ((h / l) * (-0.5 * math.pow(((M_m / 2.0) * (D / 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 (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) * sqrt(Float64(1.0 / l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D / 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 (h <= -1e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (((D / (d * (2.0 / M_m))) ^ 2.0) * -0.5))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) * sqrt((1.0 / l))) * (1.0 + ((h / l) * (-0.5 * (((M_m / 2.0) * (D / 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[h, -1e-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[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. clear-num 71.1%

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

      2. frac-times 70.3%

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

      3. *-un-lft-identity 70.3%

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

      4. *-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 82.0%

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

   8. Applied egg-rr 82.0%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

      1. pow1/2 64.7%

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

      2. div-inv 64.6%

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

      3. unpow-prod-down 72.0%

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

      4. pow1/2 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. unpow1/2 72.0%

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

   8. Simplified 72.0%

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

      1. sqrt-div 79.2%

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

   10. Applied egg-rr 79.2%

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

4. Final simplification 80.5%

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

Alternative 2: 75.6% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -1e-310)
   (*
    (sqrt (/ d h))
    (*
     (+ 1.0 (* (/ h l) (* (pow (/ D (* d (/ 2.0 M_m))) 2.0) -0.5)))
     (/ (sqrt (- d)) (sqrt (- l)))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0)) 1.0)
    (/ d (* (sqrt h) (sqrt 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) {
	double tmp;
	if (h <= -1e-310) {
		tmp = sqrt((d / h)) * ((1.0 + ((h / l) * (pow((D / (d * (2.0 / M_m))), 2.0) * -0.5))) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0) * -0.5))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	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, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $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[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. clear-num 71.1%

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

      2. frac-times 70.3%

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

      3. *-un-lft-identity 70.3%

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

      4. *-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 72.7%

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

   8. Applied egg-rr 72.7%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 75.1%

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

Alternative 3: 78.7% accurate, 0.8× speedup

Math

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

FPCore

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

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0))))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	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, -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[(-0.5 * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 82.0%

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

   6. Applied egg-rr 83.1%

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

   if -1.999999999999994e-310 < l

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 79.8%

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

Alternative 4: 78.8% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -1e-310)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (/ D (* d (/ 2.0 M_m))) 2.0) -0.5)))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0)) 1.0)
    (/ d (* (sqrt h) (sqrt 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) {
	double tmp;
	if (h <= -1e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (pow((D / (d * (2.0 / M_m))), 2.0) * -0.5))));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	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, -1e-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[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $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[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. clear-num 71.1%

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

      2. frac-times 70.3%

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

      3. *-un-lft-identity 70.3%

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

      4. *-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 82.0%

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

   8. Applied egg-rr 82.0%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 79.4%

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

Alternative 5: 73.9% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -1e-310)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (pow (* (* (/ D d) (* M_m 0.5)) (sqrt (/ h l))) 2.0))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0)) 1.0)
    (/ d (* (sqrt h) (sqrt 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) {
	double tmp;
	if (h <= -1e-310) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * pow((((D / d) * (M_m * 0.5)) * sqrt((h / l))), 2.0)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D / d) * Float64(M_m * 0.5)) * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	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, -1e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. add-sqr-sqrt 71.1%

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

      2. pow2 71.1%

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

      3. sqrt-prod 71.1%

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

      4. sqrt-pow1 73.5%

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

      5. metadata-eval 73.5%

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

      6. pow1 73.5%

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

      7. div-inv 73.5%

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

      8. metadata-eval 73.5%

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

   6. Applied egg-rr 73.5%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 75.4%

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

Alternative 6: 73.4% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -1e-310)
   (*
    (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
    (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0)))))
   (*
    (fma (/ h l) (* -0.5 (pow (* D (/ M_m (* d 2.0))) 2.0)) 1.0)
    (/ d (* (sqrt h) (sqrt 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) {
	double tmp;
	if (h <= -1e-310) {
		tmp = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M_m / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	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, -1e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $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[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 74.4%

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

Alternative 7: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 -4.3e+191)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l -1.9e-17)
     (* d (- (sqrt (/ 1.0 (* h l)))))
     (if (<= l 5.5e-303)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (/ (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0)) (/ l h))))
       (if (<= l 5.2e+147)
         (*
          (fma (pow (* D (/ M_m (* d 2.0))) 2.0) (/ (* h -0.5) l) 1.0)
          (/ d (sqrt (* h l))))
         (/ d (* (sqrt h) (sqrt 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) {
	double tmp;
	if (l <= -4.3e+191) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -1.9e-17) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 5.5e-303) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)) / (l / h)));
	} else if (l <= 5.2e+147) {
		tmp = fma(pow((D * (M_m / (d * 2.0))), 2.0), ((h * -0.5) / l), 1.0) * (d / sqrt((h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	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 <= -4.3e+191)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -1.9e-17)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 5.5e-303)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / Float64(l / h))));
	elseif (l <= 5.2e+147)
		tmp = Float64(fma((Float64(D * Float64(M_m / Float64(d * 2.0))) ^ 2.0), Float64(Float64(h * -0.5) / l), 1.0) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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, -4.3e+191], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.9e-17], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 5.5e-303], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+147], N[(N[(N[Power[N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -4.2999999999999998e191

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. pow1/2 69.7%

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

      2. div-inv 69.7%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/2 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in h around 0 62.8%

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

   if -4.2999999999999998e191 < l < -1.9000000000000001e-17

   1. Initial program 61.5%

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

   3. Simplified 64.1%

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

      1. add-sqr-sqrt 64.1%

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

      2. pow2 64.1%

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

      3. sqrt-prod 64.0%

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

      4. sqrt-pow1 66.8%

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

      5. metadata-eval 66.8%

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

      6. pow1 66.8%

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

      7. div-inv 66.8%

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

      8. metadata-eval 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. sqrt-div 66.9%

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

      3. metadata-eval 66.9%

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

   8. Applied egg-rr 66.9%

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

   9. Taylor expanded in d around -inf 54.4%

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

       1. mul-1-neg 54.4%

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

   11. Simplified 54.4%

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

   if -1.9000000000000001e-17 < l < 5.50000000000000018e-303

   1. Initial program 78.6%

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

   3. Simplified 76.9%

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

      1. associate-*r/ 80.3%

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

      2. div-inv 80.3%

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

      3. metadata-eval 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. sqrt-div 3.4%

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

   8. Applied egg-rr 3.4%

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

      1. pow1 3.4%

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

      2. sqrt-div 80.3%

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

      3. pow1/2 80.3%

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

      4. pow1/2 80.3%

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

      5. pow-prod-down 75.5%

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

      6. cancel-sign-sub-inv 75.5%

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

      7. metadata-eval 75.5%

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

      8. associate-/l* 72.1%

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

      9. *-commutative 72.1%

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

   10. Applied egg-rr 72.1%

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

       1. unpow1 72.1%

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

       2. rem-log-exp 70.6%

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

       3. rem-log-exp 72.1%

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

       4. associate-/r/ 75.5%

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

       5. associate-*l/ 75.5%

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

       6. associate-*r/ 72.1%

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

       7. *-commutative 72.1%

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

       8. unpow1/2 72.1%

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

       9. *-commutative 72.1%

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

   12. Simplified 72.1%

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

   if 5.50000000000000018e-303 < l < 5.1999999999999997e147

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   6. Applied egg-rr 82.5%

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

      1. unpow1 82.5%

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

      2. *-commutative 82.5%

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

      3. +-commutative 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*r* 82.5%

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

      6. fma-define 82.5%

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

      7. *-commutative 82.5%

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

      8. /-rgt-identity 82.5%

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

      9. associate-/l* 82.5%

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

      10. metadata-eval 82.5%

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

      11. times-frac 82.4%

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

      12. associate-*r/ 83.5%

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

      13. *-commutative 83.5%

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

   8. Simplified 83.5%

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

      1. associate-*r/ 84.7%

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

      2. associate-*r/ 83.6%

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

      3. *-commutative 83.6%

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

      4. sqrt-unprod 77.2%

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

   10. Applied egg-rr 77.2%

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

       1. associate-*r/ 76.0%

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

       2. rem-exp-log 73.6%

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

       3. fma-define 73.6%

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

       4. *-commutative 73.6%

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

       5. associate-*l* 73.6%

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

       6. fma-define 73.6%

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

       7. *-commutative 73.6%

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

       8. associate-*l/ 74.7%

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

       9. *-commutative 74.7%

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

       10. *-commutative 74.7%

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

       11. associate-*r/ 74.7%

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

       12. rem-exp-log 77.2%

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

       13. *-commutative 77.2%

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

   12. Simplified 77.2%

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

   if 5.1999999999999997e147 < l

   1. Initial program 49.3%

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

   3. Simplified 49.3%

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

   6. Applied egg-rr 65.7%

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

      1. unpow1 65.7%

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

      2. *-commutative 65.7%

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

      3. +-commutative 65.7%

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

      4. *-commutative 65.7%

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

      5. associate-*r* 65.7%

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

      6. fma-define 65.7%

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

      7. *-commutative 65.7%

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

      8. /-rgt-identity 65.7%

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

      9. associate-/l* 65.7%

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

      10. metadata-eval 65.7%

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

      11. times-frac 65.6%

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

      12. associate-*r/ 63.5%

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

      13. *-commutative 63.5%

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

   8. Simplified 63.5%

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

   9. Taylor expanded in h around 0 66.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+117}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -1.95e+191)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l -2.8e-16)
     (* d (- (sqrt (/ 1.0 (* h l)))))
     (if (<= l 2.35e-303)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (/ (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0)) (/ l h))))
       (if (<= l 3.9e+117)
         (/
          (* d (fma (/ h l) (* -0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0)) 1.0))
          (sqrt (* h l)))
         (* 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 <= -1.95e+191) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -2.8e-16) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 2.35e-303) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)) / (l / h)));
	} else if (l <= 3.9e+117) {
		tmp = (d * fma((h / l), (-0.5 * pow(((D * M_m) / (d * 2.0)), 2.0)), 1.0)) / sqrt((h * l));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.95e+191)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -2.8e-16)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 2.35e-303)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / Float64(l / h))));
	elseif (l <= 3.9e+117)
		tmp = Float64(Float64(d * fma(Float64(h / l), Float64(-0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)), 1.0)) / sqrt(Float64(h * l)));
	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_] := If[LessEqual[l, -1.95e+191], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.8e-16], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2.35e-303], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.9e+117], N[(N[(d * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.95e191

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. pow1/2 69.7%

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

      2. div-inv 69.7%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/2 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in h around 0 62.8%

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

   if -1.95e191 < l < -2.8000000000000001e-16

   1. Initial program 61.5%

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

   3. Simplified 64.1%

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

      1. add-sqr-sqrt 64.1%

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

      2. pow2 64.1%

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

      3. sqrt-prod 64.0%

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

      4. sqrt-pow1 66.8%

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

      5. metadata-eval 66.8%

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

      6. pow1 66.8%

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

      7. div-inv 66.8%

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

      8. metadata-eval 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. sqrt-div 66.9%

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

      3. metadata-eval 66.9%

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

   8. Applied egg-rr 66.9%

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

   9. Taylor expanded in d around -inf 54.4%

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

       1. mul-1-neg 54.4%

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

   11. Simplified 54.4%

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

   if -2.8000000000000001e-16 < l < 2.3499999999999999e-303

   1. Initial program 78.6%

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

   3. Simplified 76.9%

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

      1. associate-*r/ 80.3%

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

      2. div-inv 80.3%

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

      3. metadata-eval 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. sqrt-div 3.4%

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

   8. Applied egg-rr 3.4%

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

      1. pow1 3.4%

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

      2. sqrt-div 80.3%

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

      3. pow1/2 80.3%

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

      4. pow1/2 80.3%

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

      5. pow-prod-down 75.5%

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

      6. cancel-sign-sub-inv 75.5%

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

      7. metadata-eval 75.5%

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

      8. associate-/l* 72.1%

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

      9. *-commutative 72.1%

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

   10. Applied egg-rr 72.1%

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

       1. unpow1 72.1%

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

       2. rem-log-exp 70.6%

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

       3. rem-log-exp 72.1%

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

       4. associate-/r/ 75.5%

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

       5. associate-*l/ 75.5%

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

       6. associate-*r/ 72.1%

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

       7. *-commutative 72.1%

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

       8. unpow1/2 72.1%

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

       9. *-commutative 72.1%

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

   12. Simplified 72.1%

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

   if 2.3499999999999999e-303 < l < 3.8999999999999999e117

   1. Initial program 74.3%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 83.5%

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

      1. unpow1 83.5%

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

      2. *-commutative 83.5%

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

      3. +-commutative 83.5%

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

      4. *-commutative 83.5%

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

      5. associate-*r* 83.5%

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

      6. fma-define 83.5%

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

      7. *-commutative 83.5%

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

      8. /-rgt-identity 83.5%

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

      9. associate-/l* 83.5%

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

      10. metadata-eval 83.5%

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

      11. times-frac 84.6%

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

      12. associate-*r/ 84.6%

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

      13. *-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. associate-*r/ 85.9%

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

      2. associate-*r/ 85.9%

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

      3. *-commutative 85.9%

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

      4. sqrt-unprod 81.4%

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

   10. Applied egg-rr 81.4%

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

   if 3.8999999999999999e117 < l

   1. Initial program 47.7%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in d around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. associate-/r* 49.0%

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

   7. Simplified 49.0%

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

      1. *-un-lft-identity 49.0%

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

      2. sqrt-div 62.3%

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

      3. pow1/2 62.3%

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

      4. inv-pow 62.3%

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

      5. pow-pow 62.3%

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

      6. metadata-eval 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. *-lft-identity 62.3%

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

   11. Simplified 62.3%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+117}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+260}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M\_m \cdot \frac{D \cdot 0.5}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 -1.1e+260)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= d -9.6e-288)
     (*
      (fma (/ h l) (* -0.5 (pow (* M_m (/ (* D 0.5) d)) 2.0)) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M_m) (* d 2.0)) 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 <= -1.1e+260) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (d <= -9.6e-288) {
		tmp = fma((h / l), (-0.5 * pow((M_m * ((D * 0.5) / d)), 2.0)), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / l) * (-0.5 * pow(((D * M_m) / (d * 2.0)), 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 <= -1.1e+260)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -9.6e-288)
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(M_m * Float64(Float64(D * 0.5) / d)) ^ 2.0)), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 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, -1.1e+260], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -9.6e-288], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10000000000000006e260

   1. Initial program 60.5%

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

   3. Simplified 59.7%

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

      1. add-sqr-sqrt 59.7%

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

      2. pow2 59.7%

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

      3. sqrt-prod 59.6%

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

      4. sqrt-pow1 59.6%

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

      5. metadata-eval 59.6%

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

      6. pow1 59.6%

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

      7. div-inv 59.6%

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

      8. metadata-eval 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. clear-num 59.8%

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

      2. sqrt-div 59.7%

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

      3. metadata-eval 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in d around -inf 83.4%

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

       1. mul-1-neg 83.4%

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

   11. Simplified 83.4%

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

   if -1.10000000000000006e260 < d < -9.5999999999999994e-288

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. add-sqr-sqrt 74.5%

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

      2. pow2 74.5%

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

      3. sqrt-prod 74.5%

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

      4. sqrt-pow1 77.3%

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

      5. metadata-eval 77.3%

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

      6. pow1 77.3%

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

      7. div-inv 77.3%

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

      8. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. pow1 77.3%

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

   8. Applied egg-rr 66.2%

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

      1. unpow1 66.2%

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

      2. *-commutative 66.2%

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

      3. +-commutative 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*r* 66.2%

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

      6. fma-undefine 66.2%

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

      7. *-commutative 66.2%

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

      8. associate-*r/ 66.2%

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

   10. Simplified 66.2%

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

   if -9.5999999999999994e-288 < d

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   6. Applied egg-rr 75.5%

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

      1. unpow1 75.5%

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

      2. *-commutative 75.5%

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

      3. +-commutative 75.5%

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

      4. *-commutative 75.5%

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

      5. associate-*r* 75.5%

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

      6. fma-define 75.5%

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

      7. *-commutative 75.5%

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

      8. /-rgt-identity 75.5%

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

      9. associate-/l* 75.5%

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

      10. metadata-eval 75.5%

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

      11. times-frac 75.4%

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

      12. associate-*r/ 75.4%

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

      13. *-commutative 75.4%

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

   8. Simplified 75.4%

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

      1. fma-undefine 75.4%

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

      2. associate-*r/ 75.4%

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

      3. *-commutative 75.4%

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

   10. Applied egg-rr 75.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+260}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 1.0× speedup

Math

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

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 M_m) (* d 2.0)) 2.0)))
   (if (<= h -1e-310)
     (*
      (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
      (- 1.0 (* (/ h l) (* 0.5 t_0))))
     (* (/ 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 * M_m) / (d * 2.0)), 2.0);
	double tmp;
	if (h <= -1e-310) {
		tmp = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * t_0)));
	} 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 * m_m) / (d * 2.0d0)) ** 2.0d0
    if (h <= (-1d-310)) then
        tmp = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * t_0)))
    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 * M_m) / (d * 2.0)), 2.0);
	double tmp;
	if (h <= -1e-310) {
		tmp = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * t_0)));
	} 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 * M_m) / (d * 2.0)), 2.0)
	tmp = 0
	if h <= -1e-310:
		tmp = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * t_0)))
	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(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))));
	else
		tmp = Float64(Float64(d / Float64(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 * M_m) / (d * 2.0)) ^ 2.0;
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * t_0)));
	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[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -1e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $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 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

      1. fma-undefine 77.1%

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

      2. associate-*r/ 77.1%

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

      3. *-commutative 77.1%

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

   10. Applied egg-rr 77.1%

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

4. Final simplification 74.4%

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

Alternative 11: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-303}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 2.35e-303)
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
    (sqrt (/ d h)))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M_m) (* d 2.0)) 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 <= 2.35e-303) {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow(((M_m / 2.0) * (D / d)), 2.0))))) * sqrt((d / h));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / l) * (-0.5 * pow(((D * M_m) / (d * 2.0)), 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 <= 2.35d-303) then
        tmp = (sqrt((d / l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))) * sqrt((d / h))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 * m_m) / (d * 2.0d0)) ** 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 <= 2.35e-303) {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((M_m / 2.0) * (D / d)), 2.0))))) * Math.sqrt((d / h));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D * M_m) / (d * 2.0)), 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 <= 2.35e-303:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((M_m / 2.0) * (D / d)), 2.0))))) * math.sqrt((d / h))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + ((h / l) * (-0.5 * math.pow(((D * M_m) / (d * 2.0)), 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 <= 2.35e-303)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0))))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 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 <= 2.35e-303)
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * (((M_m / 2.0) * (D / d)) ^ 2.0))))) * sqrt((d / h));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / l) * (-0.5 * (((D * M_m) / (d * 2.0)) ^ 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, 2.35e-303], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3499999999999999e-303

   1. Initial program 71.7%

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

   3. Simplified 71.6%

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

   if 2.3499999999999999e-303 < l

   1. Initial program 64.1%

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

   3. Simplified 64.2%

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

   6. Applied egg-rr 76.8%

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

      1. unpow1 76.8%

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

      2. *-commutative 76.8%

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

      3. +-commutative 76.8%

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

      4. *-commutative 76.8%

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

      5. associate-*r* 76.8%

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

      6. fma-define 76.8%

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

      7. *-commutative 76.8%

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

      8. /-rgt-identity 76.8%

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

      9. associate-/l* 76.8%

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

      10. metadata-eval 76.8%

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

      11. times-frac 76.7%

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

      12. associate-*r/ 76.8%

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

      13. *-commutative 76.8%

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

   8. Simplified 76.8%

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

      1. fma-undefine 76.8%

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

      2. associate-*r/ 76.7%

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

      3. *-commutative 76.7%

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

   10. Applied egg-rr 76.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-303}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 1.0× speedup

Math

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

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 (<= h -1e-310)
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (/ D (* d (/ 2.0 M_m))) 2.0) -0.5))))
    (sqrt (/ d h)))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M_m) (* d 2.0)) 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 (h <= -1e-310) {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (pow((D / (d * (2.0 / M_m))), 2.0) * -0.5)))) * sqrt((d / h));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / l) * (-0.5 * pow(((D * M_m) / (d * 2.0)), 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 (h <= (-1d-310)) then
        tmp = (sqrt((d / l)) * (1.0d0 + ((h / l) * (((d_1 / (d * (2.0d0 / m_m))) ** 2.0d0) * (-0.5d0))))) * sqrt((d / h))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 * m_m) / (d * 2.0d0)) ** 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 (h <= -1e-310) {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h / l) * (Math.pow((D / (d * (2.0 / M_m))), 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 * M_m) / (d * 2.0)), 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 h <= -1e-310:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h / l) * (math.pow((D / (d * (2.0 / M_m))), 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 * M_m) / (d * 2.0)), 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 (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0) * -0.5)))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 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 (h <= -1e-310)
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (((D / (d * (2.0 / M_m))) ^ 2.0) * -0.5)))) * sqrt((d / h));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / l) * (-0.5 * (((D * M_m) / (d * 2.0)) ^ 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[h, -1e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 71.2%

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

   3. Simplified 71.1%

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

      1. clear-num 71.1%

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

      2. frac-times 70.3%

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

      3. *-un-lft-identity 70.3%

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

      4. *-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

   if -9.999999999999969e-311 < h

   1. Initial program 64.6%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 77.1%

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

      1. unpow1 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*r* 77.1%

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

      6. fma-define 77.1%

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

      7. *-commutative 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. associate-/l* 77.1%

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

      10. metadata-eval 77.1%

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

      11. times-frac 77.1%

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

      12. associate-*r/ 77.1%

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

      13. *-commutative 77.1%

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

   8. Simplified 77.1%

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

      1. fma-undefine 77.1%

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

      2. associate-*r/ 77.1%

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

      3. *-commutative 77.1%

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

   10. Applied egg-rr 77.1%

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

4. Final simplification 73.9%

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

Alternative 13: 57.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.45 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

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 2.45e-14)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (/ (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0)) (/ l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (D <= 2.45e-14) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)) / (l / 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 (d_1 <= 2.45d-14) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + (((-0.5d0) * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / (l / 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 (D <= 2.45e-14) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / (l / h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if D <= 2.45e-14:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / (l / h)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (D <= 2.45e-14)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / Float64(l / 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 (D <= 2.45e-14)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((-0.5 * (((D / d) * (M_m * 0.5)) ^ 2.0)) / (l / 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[D, 2.45e-14], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.44999999999999997e-14

   1. Initial program 67.8%

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

   3. Simplified 67.3%

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

      1. pow1/2 67.3%

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

      2. div-inv 67.3%

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

      3. unpow-prod-down 41.8%

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

      4. pow1/2 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. unpow1/2 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in h around 0 45.4%

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

   if 2.44999999999999997e-14 < D

   1. Initial program 67.3%

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

   3. Simplified 68.5%

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

      1. associate-*r/ 67.3%

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

      2. div-inv 67.3%

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

      3. metadata-eval 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. sqrt-div 34.2%

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

   8. Applied egg-rr 30.1%

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

      1. pow1 30.1%

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

      2. sqrt-div 67.3%

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

      3. pow1/2 67.3%

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

      4. pow1/2 67.3%

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

      5. pow-prod-down 62.0%

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

      6. cancel-sign-sub-inv 62.0%

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

      7. metadata-eval 62.0%

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

      8. associate-/l* 63.1%

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

      9. *-commutative 63.1%

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

   10. Applied egg-rr 63.1%

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

       1. unpow1 63.1%

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

       2. rem-log-exp 60.6%

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

       3. rem-log-exp 63.1%

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

       4. associate-/r/ 64.5%

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

       5. associate-*l/ 62.0%

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

       6. associate-*r/ 63.2%

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

       7. *-commutative 63.2%

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

       8. unpow1/2 63.2%

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

       9. *-commutative 63.2%

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

   12. Simplified 63.1%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.45 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -7e+198)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l 1.75e-169)
     (* d (- (pow (* 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 <= -7e+198) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= 1.75e-169) {
		tmp = d * -pow((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 <= (-7d+198)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= 1.75d-169) then
        tmp = d * -((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 <= -7e+198) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= 1.75e-169) {
		tmp = d * -Math.pow((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 <= -7e+198:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= 1.75e-169:
		tmp = d * -math.pow((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 <= -7e+198)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= 1.75e-169)
		tmp = Float64(d * 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 <= -7e+198)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= 1.75e-169)
		tmp = d * -((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, -7e+198], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e-169], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $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 < -7.00000000000000026e198

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

      1. pow1/2 72.5%

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

      2. div-inv 72.5%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/2 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in h around 0 65.2%

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

   if -7.00000000000000026e198 < l < 1.7500000000000001e-169

   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 72.3%

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

      1. add-sqr-sqrt 72.3%

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

      2. pow2 72.3%

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

      3. sqrt-prod 72.3%

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

      4. sqrt-pow1 74.6%

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

      5. metadata-eval 74.6%

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

      6. pow1 74.6%

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

      7. div-inv 74.6%

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

      8. metadata-eval 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. clear-num 74.6%

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

      2. sqrt-div 74.6%

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

      3. metadata-eval 74.6%

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

   8. Applied egg-rr 74.6%

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

   9. Taylor expanded in d around -inf 46.3%

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

       1. associate-*r* 46.3%

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

       2. mul-1-neg 46.3%

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

       3. *-commutative 46.3%

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

       4. rem-exp-log 44.8%

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

       5. rec-exp 44.8%

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

       6. unpow1/2 44.8%

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

       7. exp-prod 45.9%

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

       8. *-commutative 45.9%

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

       9. distribute-rgt-neg-out 45.9%

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

       10. distribute-lft-neg-in 45.9%

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

       11. metadata-eval 45.9%

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

       12. *-commutative 45.9%

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

       13. exp-to-pow 47.5%

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

       14. *-commutative 47.5%

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

   11. Simplified 47.5%

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

   if 1.7500000000000001e-169 < l

   1. Initial program 61.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-/r* 48.2%

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

   7. Simplified 48.2%

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

      1. *-un-lft-identity 48.2%

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

      2. sqrt-div 55.5%

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

      3. pow1/2 55.5%

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

      4. inv-pow 55.5%

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

      5. pow-pow 55.6%

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

      6. metadata-eval 55.6%

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

   9. Applied egg-rr 55.6%

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

       1. *-lft-identity 55.6%

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

   11. Simplified 55.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 2.6e-169)
   (* d (- (pow (* 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 <= 2.6e-169) {
		tmp = d * -pow((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 <= 2.6d-169) then
        tmp = d * -((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 <= 2.6e-169) {
		tmp = d * -Math.pow((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 <= 2.6e-169:
		tmp = d * -math.pow((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 <= 2.6e-169)
		tmp = Float64(d * 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 <= 2.6e-169)
		tmp = d * -((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, 2.6e-169], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.60000000000000014e-169

   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 72.3%

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

      1. add-sqr-sqrt 72.3%

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

      2. pow2 72.3%

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

      3. sqrt-prod 72.3%

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

      4. sqrt-pow1 74.9%

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

      5. metadata-eval 74.9%

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

      6. pow1 74.9%

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

      7. div-inv 74.9%

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

      8. metadata-eval 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. clear-num 74.3%

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

      2. sqrt-div 74.3%

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

      3. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

   9. Taylor expanded in d around -inf 46.3%

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

       1. associate-*r* 46.3%

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

       2. mul-1-neg 46.3%

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

       3. *-commutative 46.3%

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

       4. rem-exp-log 44.6%

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

       5. rec-exp 44.6%

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

       6. unpow1/2 44.6%

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

       7. exp-prod 45.5%

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

       8. *-commutative 45.5%

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

       9. distribute-rgt-neg-out 45.5%

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

       10. distribute-lft-neg-in 45.5%

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

       11. metadata-eval 45.5%

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

       12. *-commutative 45.5%

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

       13. exp-to-pow 47.2%

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

       14. *-commutative 47.2%

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

   11. Simplified 47.2%

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

   if 2.60000000000000014e-169 < l

   1. Initial program 61.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-/r* 48.2%

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

   7. Simplified 48.2%

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

      1. *-un-lft-identity 48.2%

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

      2. sqrt-div 55.5%

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

      3. pow1/2 55.5%

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

      4. inv-pow 55.5%

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

      5. pow-pow 55.6%

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

      6. metadata-eval 55.6%

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

   9. Applied egg-rr 55.6%

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

       1. *-lft-identity 55.6%

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

   11. Simplified 55.6%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 1.8e-169)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.8e-169) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / 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 <= 1.8d-169) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / 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 <= 1.8e-169) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / 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 <= 1.8e-169:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 1.8e-169)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / 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 <= 1.8e-169)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / 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, 1.8e-169], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.80000000000000001e-169

   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 72.3%

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

      1. add-sqr-sqrt 72.3%

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

      2. pow2 72.3%

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

      3. sqrt-prod 72.3%

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

      4. sqrt-pow1 74.9%

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

      5. metadata-eval 74.9%

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

      6. pow1 74.9%

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

      7. div-inv 74.9%

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

      8. metadata-eval 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. clear-num 74.3%

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

      2. sqrt-div 74.3%

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

      3. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

   9. Taylor expanded in d around -inf 46.3%

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

       1. associate-*r* 46.3%

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

       2. mul-1-neg 46.3%

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

       3. *-commutative 46.3%

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

       4. rem-exp-log 44.6%

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

       5. rec-exp 44.6%

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

       6. unpow1/2 44.6%

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

       7. exp-prod 45.5%

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

       8. *-commutative 45.5%

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

       9. distribute-rgt-neg-out 45.5%

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

       10. distribute-lft-neg-in 45.5%

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

       11. metadata-eval 45.5%

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

       12. *-commutative 45.5%

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

       13. exp-to-pow 47.2%

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

       14. *-commutative 47.2%

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

   11. Simplified 47.2%

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

   if 1.80000000000000001e-169 < l

   1. Initial program 61.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-/r* 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in d around inf 26.6%

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

   1. *-commutative 26.6%

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

   2. associate-/r* 26.8%

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

7. Simplified 26.8%

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

8. Final simplification 26.8%

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

Alternative 18: 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 67.7%

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

3. Simplified 67.7%

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

   1. add-sqr-sqrt 67.6%

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

   2. pow2 67.6%

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

   3. sqrt-prod 67.6%

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

   4. sqrt-pow1 70.4%

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

   5. metadata-eval 70.4%

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

   6. pow1 70.4%

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

   7. div-inv 70.4%

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

   8. metadata-eval 70.4%

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

6. Applied egg-rr 70.4%

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

7. Taylor expanded in d around inf 26.6%

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

   1. unpow-1 26.6%

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

   2. sqr-pow 26.6%

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

   3. rem-sqrt-square 26.6%

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

   4. metadata-eval 26.6%

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

   5. sqr-pow 26.5%

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

   6. fabs-sqr 26.5%

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

   7. sqr-pow 26.6%

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

9. Simplified 26.6%

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

10. Final simplification 26.6%

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

Reproduce

herbie shell --seed 2024039 
(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)))))