Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 79.8%

Time: 22.9s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

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 (sqrt (- d)))
        (t_1 (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M_m 2.0) (/ D d)) 2.0))))))
   (if (<= l -3e+217)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (fma (/ h l) (* -0.5 (pow (/ D (* d (/ 2.0 M_m))) 2.0)) 1.0)))
     (if (<= l -1e-310)
       (* (/ t_0 (sqrt (- h))) (* (sqrt (/ d l)) t_1))
       (* (/ (sqrt d) (sqrt h)) (* t_1 (/ (sqrt d) (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 t_0 = sqrt(-d);
	double t_1 = 1.0 + ((h / l) * (-0.5 * pow(((M_m / 2.0) * (D / d)), 2.0)));
	double tmp;
	if (l <= -3e+217) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * fma((h / l), (-0.5 * pow((D / (d * (2.0 / M_m))), 2.0)), 1.0));
	} else if (l <= -1e-310) {
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * t_1);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (sqrt(d) / 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)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0))))
	tmp = 0.0
	if (l <= -3e+217)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * fma(Float64(h / l), Float64(-0.5 * (Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0)), 1.0)));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_1));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(sqrt(d) / 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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[l, -3e+217], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.99999999999999976e217

   1. Initial program 57.7%

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

   3. Simplified 48.1%

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

      1. frac-2neg 48.1%

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

      2. sqrt-div 60.8%

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

   6. Applied egg-rr 60.8%

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

   if -2.99999999999999976e217 < l < -9.999999999999969e-311

   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%

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

      1. frac-2neg 71.7%

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

      2. sqrt-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   3. Simplified 68.2%

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

      1. sqrt-div 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. sqrt-div 83.1%

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

   8. Applied egg-rr 83.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M\_m}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -1.3e+218)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (fma (/ h l) (* -0.5 (pow (/ D (* d (/ 2.0 M_m))) 2.0)) 1.0)))
     (if (<= l -1e-310)
       (*
        (/ t_0 (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 (/ M_m (/ d (* D 0.5))) 2.0)) 1.0)
        (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -1.3e+218) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * fma((h / l), (-0.5 * pow((D / (d * (2.0 / M_m))), 2.0)), 1.0));
	} else if (l <= -1e-310) {
		tmp = (t_0 / 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((M_m / (d / (D * 0.5))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.3e+218)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * fma(Float64(h / l), Float64(-0.5 * (Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0)), 1.0)));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(t_0 / 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(M_m / Float64(d / Float64(D * 0.5))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.3e+218], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(t$95$0 / 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[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.30000000000000001e218

   1. Initial program 57.7%

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

   3. Simplified 48.1%

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

      1. frac-2neg 48.1%

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

      2. sqrt-div 60.8%

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

   6. Applied egg-rr 60.8%

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

   if -1.30000000000000001e218 < l < -9.999999999999969e-311

   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%

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

      1. frac-2neg 71.7%

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

      2. sqrt-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M\_m}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M\_m}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -3.3e+216)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ D (/ 2.0 M_m)) d) 2.0))))))
     (if (<= l -1e-310)
       (*
        (* (/ t_0 (sqrt (- h))) (sqrt (/ d l)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
       (*
        (fma (/ h l) (* -0.5 (pow (/ M_m (/ d (* D 0.5))) 2.0)) 1.0)
        (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -3.3e+216) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / (2.0 / M_m)) / d), 2.0)))));
	} else if (l <= -1e-310) {
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = fma((h / l), (-0.5 * pow((M_m / (d / (D * 0.5))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.3e+216)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / Float64(2.0 / M_m)) / d) ^ 2.0))))));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(M_m / Float64(d / Float64(D * 0.5))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.3e+216], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.3e216

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

      1. clear-num 57.8%

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

      2. frac-times 48.1%

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

      3. *-un-lft-identity 48.1%

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

      4. associate-/r* 57.7%

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

   6. Applied egg-rr 57.7%

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

      1. frac-2neg 48.1%

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

      2. sqrt-div 60.8%

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

   8. Applied egg-rr 71.5%

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

   if -3.3e216 < l < -9.999999999999969e-311

   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
   5. Step-by-step derivation

      1. frac-2neg 71.7%

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

      2. sqrt-div 86.9%

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

   6. Applied egg-rr 83.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M\_m}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M\_m}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -6.8e+215)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ D (/ 2.0 M_m)) d) 2.0))))))
     (if (<= l -1e-310)
       (*
        (/ t_0 (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 (/ M_m (/ d (* D 0.5))) 2.0)) 1.0)
        (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -6.8e+215) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / (2.0 / M_m)) / d), 2.0)))));
	} else if (l <= -1e-310) {
		tmp = (t_0 / 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((M_m / (d / (D * 0.5))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -6.8e+215)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / Float64(2.0 / M_m)) / d) ^ 2.0))))));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(t_0 / 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(M_m / Float64(d / Float64(D * 0.5))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -6.8e+215], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(t$95$0 / 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[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.80000000000000036e215

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

      1. clear-num 57.8%

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

      2. frac-times 48.1%

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

      3. *-un-lft-identity 48.1%

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

      4. associate-/r* 57.7%

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

   6. Applied egg-rr 57.7%

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

      1. frac-2neg 48.1%

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

      2. sqrt-div 60.8%

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

   8. Applied egg-rr 71.5%

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

   if -6.80000000000000036e215 < l < -9.999999999999969e-311

   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%

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

      1. frac-2neg 71.7%

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

      2. sqrt-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-310) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / (2.0 / M_m)) / d), 2.0)))));
	} else {
		tmp = fma((h / l), (-0.5 * pow((M_m / (d / (D * 0.5))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Simplified 69.2%

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

      1. clear-num 69.2%

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

      2. frac-times 66.8%

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

      3. *-un-lft-identity 66.8%

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

      4. associate-/r* 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. frac-2neg 66.8%

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

      2. sqrt-div 69.8%

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

   8. Applied egg-rr 72.4%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 76.9%

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

Alternative 6: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M\_m}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -2.7e+91)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l 3e-307)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (/ (* h (* -0.5 (pow (/ D (* d (/ 2.0 M_m))) 2.0))) l))))
       (*
        (fma (/ h l) (* -0.5 (pow (/ M_m (/ d (* D 0.5))) 2.0)) 1.0)
        (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -2.7e+91) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= 3e-307) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * pow((D / (d * (2.0 / M_m))), 2.0))) / l)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((M_m / (d / (D * 0.5))), 2.0)), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.7e+91)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= 3e-307)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0))) / l))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(M_m / Float64(d / Float64(D * 0.5))) ^ 2.0)), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.7e+91], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 3e-307], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.7e91

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -2.7e91 < l < 2.9999999999999999e-307

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

      1. clear-num 79.8%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. associate-/r* 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*l/ 81.1%

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

      2. associate-/l/ 80.0%

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

   8. Applied egg-rr 80.0%

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

   if 2.9999999999999999e-307 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -4.7e+91)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l -1e-310)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M_m 2.0) (/ D d)) 2.0))))))
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 t_0 = sqrt((d / l));
	double tmp;
	if (l <= -4.7e+91) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * pow(((M_m / 2.0) * (D / d)), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-4.7d+91)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= (-1d-310)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -4.7e+91) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * Math.pow(((M_m / 2.0) * (D / d)), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -4.7e+91:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= -1e-310:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * math.pow(((M_m / 2.0) * (D / d)), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4.7e+91)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -4.7e+91)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= -1e-310)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * (((M_m / 2.0) * (D / d)) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.7e+91], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * 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[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.6999999999999997e91

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -4.6999999999999997e91 < l < -9.999999999999969e-311

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{\frac{d}{D \cdot 0.5}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -5.2e+91)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l -1e-310)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (/ M_m (/ d (* D 0.5))) 2.0))))))
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 t_0 = sqrt((d / l));
	double tmp;
	if (l <= -5.2e+91) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * pow((M_m / (d / (D * 0.5))), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-5.2d+91)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= (-1d-310)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * ((m_m / (d / (d_1 * 0.5d0))) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -5.2e+91) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * Math.pow((M_m / (d / (D * 0.5))), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -5.2e+91:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= -1e-310:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * math.pow((M_m / (d / (D * 0.5))), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5.2e+91)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(M_m / Float64(d / Float64(D * 0.5))) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -5.2e+91)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= -1e-310)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * ((M_m / (d / (D * 0.5))) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.2e+91], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.2000000000000001e91

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -5.2000000000000001e91 < l < -9.999999999999969e-311

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in M around 0 79.9%

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

      1. associate-*r/ 79.9%

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

      2. associate-*r* 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/l* 79.9%

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

      5. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.92 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M\_m}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -1.92e+84)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l -1e-310)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ D (/ 2.0 M_m)) d) 2.0))))))
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 t_0 = sqrt((d / l));
	double tmp;
	if (l <= -1.92e+84) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * pow(((D / (2.0 / M_m)) / d), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-1.92d+84)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= (-1d-310)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / (2.0d0 / m_m)) / d) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.92e+84) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * Math.pow(((D / (2.0 / M_m)) / d), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.92e+84:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= -1e-310:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * math.pow(((D / (2.0 / M_m)) / d), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.92e+84)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / Float64(2.0 / M_m)) / d) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.92e+84)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= -1e-310)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * (((D / (2.0 / M_m)) / d) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.92e+84], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.91999999999999993e84

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -1.91999999999999993e84 < l < -9.999999999999969e-311

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

      1. clear-num 79.8%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. associate-/r* 79.8%

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

   6. Applied egg-rr 79.8%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.92 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D}{\frac{2}{M}}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D \cdot M\_m}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -5.8e+90)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l -1e-310)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ (* D M_m) 2.0) d) 2.0))))))
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 t_0 = sqrt((d / l));
	double tmp;
	if (l <= -5.8e+90) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * pow((((D * M_m) / 2.0) / d), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-5.8d+90)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= (-1d-310)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * ((((d_1 * m_m) / 2.0d0) / d) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -5.8e+90) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= -1e-310) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * Math.pow((((D * M_m) / 2.0) / d), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -5.8e+90:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= -1e-310:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * math.pow((((D * M_m) / 2.0) / d), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5.8e+90)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(Float64(D * M_m) / 2.0) / d) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -5.8e+90)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= -1e-310)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h / l) * (-0.5 * ((((D * M_m) / 2.0) / d) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.8e+90], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(D * M$95$m), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.8000000000000003e90

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -5.8000000000000003e90 < l < -9.999999999999969e-311

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

      1. frac-times 79.9%

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

      2. associate-/r* 79.9%

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

   6. Applied egg-rr 79.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_1 \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot t\_0\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

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 d) (* M_m 0.5)) 2.0)) (t_1 (sqrt (/ d l))))
   (if (<= l -6.2e+90)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_1)
     (if (<= l 5e-308)
       (* (sqrt (/ d h)) (* t_1 (+ 1.0 (/ (* h (* -0.5 t_0)) l))))
       (* (/ (/ d (sqrt l)) (sqrt h)) (- 1.0 (* 0.5 (* (/ h l) 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 / d) * (M_m * 0.5)), 2.0);
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -6.2e+90) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_1;
	} else if (l <= 5e-308) {
		tmp = sqrt((d / h)) * (t_1 * (1.0 + ((h * (-0.5 * t_0)) / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * 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) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0
    t_1 = sqrt((d / l))
    if (l <= (-6.2d+90)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_1
    else if (l <= 5d-308) then
        tmp = sqrt((d / h)) * (t_1 * (1.0d0 + ((h * ((-0.5d0) * t_0)) / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * 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 / d) * (M_m * 0.5)), 2.0);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= -6.2e+90) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_1;
	} else if (l <= 5e-308) {
		tmp = Math.sqrt((d / h)) * (t_1 * (1.0 + ((h * (-0.5 * t_0)) / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * 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 / d) * (M_m * 0.5)), 2.0)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= -6.2e+90:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_1
	elif l <= 5e-308:
		tmp = math.sqrt((d / h)) * (t_1 * (1.0 + ((h * (-0.5 * t_0)) / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * 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 / d) * Float64(M_m * 0.5)) ^ 2.0
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -6.2e+90)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_1);
	elseif (l <= 5e-308)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_1 * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * t_0)) / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * 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 / d) * (M_m * 0.5)) ^ 2.0;
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -6.2e+90)
		tmp = (sqrt(-d) / sqrt(-h)) * t_1;
	elseif (l <= 5e-308)
		tmp = sqrt((d / h)) * (t_1 * (1.0 + ((h * (-0.5 * t_0)) / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * 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 / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6.2e+90], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 5e-308], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(N[(h * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.19999999999999977e90

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -6.19999999999999977e90 < l < 4.99999999999999955e-308

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

      1. associate-*l/ 81.1%

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

      2. div-inv 81.1%

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

      3. metadata-eval 81.1%

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

   6. Applied egg-rr 81.1%

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

   if 4.99999999999999955e-308 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -1.85e+89)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= l 1.9e-308)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (/ (* h (* -0.5 (pow (/ D (* d (/ 2.0 M_m))) 2.0))) l))))
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 t_0 = sqrt((d / l));
	double tmp;
	if (l <= -1.85e+89) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (l <= 1.9e-308) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * pow((D / (d * (2.0 / M_m))), 2.0))) / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-1.85d+89)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else if (l <= 1.9d-308) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h * ((-0.5d0) * ((d_1 / (d * (2.0d0 / m_m))) ** 2.0d0))) / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.85e+89) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else if (l <= 1.9e-308) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * Math.pow((D / (d * (2.0 / M_m))), 2.0))) / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.85e+89:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	elif l <= 1.9e-308:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * math.pow((D / (d * (2.0 / M_m))), 2.0))) / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.85e+89)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (l <= 1.9e-308)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 2.0))) / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.85e+89)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	elseif (l <= 1.9e-308)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * ((D / (d * (2.0 / M_m))) ^ 2.0))) / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.85e+89], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 1.9e-308], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8499999999999999e89

   1. Initial program 48.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in h around 0 38.8%

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

      1. frac-2neg 47.9%

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

      2. sqrt-div 65.2%

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

   7. Applied egg-rr 58.4%

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

   if -1.8499999999999999e89 < l < 1.9000000000000001e-308

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

      1. clear-num 79.8%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. associate-/r* 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*l/ 81.1%

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

      2. associate-/l/ 80.0%

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

   8. Applied egg-rr 80.0%

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

   if 1.9000000000000001e-308 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.6% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1e-310)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (* M_m 0.5)) 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 <= -1e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m * 0.5)), 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 <= (-1d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m * 0.5d0)) ** 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 <= -1e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m * 0.5)), 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 <= -1e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m * 0.5)), 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 <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 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 <= -1e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m * 0.5)) ^ 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, -1e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in h around 0 34.4%

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

      1. frac-2neg 69.2%

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

      2. sqrt-div 82.6%

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

   7. Applied egg-rr 43.8%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   4. Applied egg-rr 26.4%

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

      1. expm1-def 41.0%

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

      2. expm1-log1p 80.9%

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

      3. associate-*r* 80.9%

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

      4. associate-/r* 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 63.5%

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

Alternative 14: 47.1% accurate, 1.1× speedup

Math

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

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 -1e-310)
   (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- 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 <= -1e-310) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-310)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-310) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1e-310:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	else:
		tmp = d / (math.sqrt(h) * math.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 <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in h around 0 34.4%

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

      1. frac-2neg 66.8%

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

      2. sqrt-div 69.8%

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

   7. Applied egg-rr 37.5%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in h around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. sqrt-div 37.6%

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

      3. sqrt-div 42.3%

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

      4. frac-times 42.3%

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

      5. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 1.1× speedup

Math

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

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 -1e-310)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d 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 <= -1e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))
	else:
		tmp = d / (math.sqrt(h) * math.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 <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in h around 0 34.4%

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

      1. frac-2neg 69.2%

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

      2. sqrt-div 82.6%

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

   7. Applied egg-rr 43.8%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in h around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. sqrt-div 37.6%

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

      3. sqrt-div 42.3%

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

      4. frac-times 42.3%

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

      5. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= d -2.3e-175)
   (sqrt (/ (pow d 2.0) (* l h)))
   (if (<= d -4e-310)
     (* d (sqrt (/ 1.0 (* l h))))
     (/ 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 (d <= -2.3e-175) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else if (d <= -4e-310) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	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 <= (-2.3d-175)) then
        tmp = sqrt(((d ** 2.0d0) / (l * h)))
    else if (d <= (-4d-310)) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    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 <= -2.3e-175) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else if (d <= -4e-310) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	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 <= -2.3e-175:
		tmp = math.sqrt((math.pow(d, 2.0) / (l * h)))
	elif d <= -4e-310:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.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 (d <= -2.3e-175)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	elseif (d <= -4e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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 <= -2.3e-175)
		tmp = sqrt(((d ^ 2.0) / (l * h)));
	elseif (d <= -4e-310)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	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, -2.3e-175], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -4e-310], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3e-175

   1. Initial program 77.1%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in h around 0 41.2%

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

      1. sqrt-unprod 37.1%

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

      2. pow1/2 37.1%

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

      3. frac-times 30.5%

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

      4. pow2 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. unpow1/2 30.5%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   9. Simplified 30.5%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -2.3e-175 < d < -3.999999999999988e-310

   1. Initial program 46.8%

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

   3. Taylor expanded in d around inf 21.9%

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

   if -3.999999999999988e-310 < d

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in h around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. sqrt-div 37.6%

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

      3. sqrt-div 42.3%

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

      4. frac-times 42.3%

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

      5. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 5.1e-212)
   (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
   (/ 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 <= 5.1e-212) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	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 <= 5.1d-212) then
        tmp = sqrt((d / l)) * (1.0d0 / sqrt((h / d)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    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 <= 5.1e-212) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	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 <= 5.1e-212:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = d / (math.sqrt(h) * math.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 <= 5.1e-212)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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 <= 5.1e-212)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	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, 5.1e-212], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.0999999999999997e-212

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. clear-num 70.0%

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

      2. sqrt-div 71.7%

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

      3. metadata-eval 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in h around 0 36.0%

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

   if 5.0999999999999997e-212 < l

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in h around 0 34.3%

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

      1. *-commutative 34.3%

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

      2. sqrt-div 38.4%

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

      3. sqrt-div 44.1%

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

      4. frac-times 44.1%

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

      5. add-sqr-sqrt 44.3%

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

   7. Applied egg-rr 44.3%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 2.25e-212)
   (* (sqrt (/ d h)) (sqrt (/ d 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 <= 2.25e-212) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	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 <= 2.25d-212) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    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 <= 2.25e-212) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	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 <= 2.25e-212:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = d / (math.sqrt(h) * math.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 <= 2.25e-212)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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 <= 2.25e-212)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	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, 2.25e-212], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2499999999999999e-212

   1. Initial program 70.9%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in h around 0 34.3%

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

   if 2.2499999999999999e-212 < l

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in h around 0 34.3%

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

      1. *-commutative 34.3%

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

      2. sqrt-div 38.4%

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

      3. sqrt-div 44.1%

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

      4. frac-times 44.1%

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

      5. add-sqr-sqrt 44.3%

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

   7. Applied egg-rr 44.3%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.7% accurate, 1.6× speedup

Math

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

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 -1e-310) (* d (sqrt (/ 1.0 (* l h)))) (/ 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 <= -1e-310) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1e-310:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.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 <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Taylor expanded in d around inf 11.2%

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

   if -9.999999999999969e-311 < l

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in h around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. sqrt-div 37.6%

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

      3. sqrt-div 42.3%

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

      4. frac-times 42.3%

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

      5. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

3. Taylor expanded in d around inf 25.3%

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

4. Final simplification 25.3%

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

Alternative 21: 27.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

3. Simplified 67.9%

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

5. Taylor expanded in h around 0 34.3%

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

6. Taylor expanded in d around 0 25.3%

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

   1. associate-/r* 26.0%

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

8. Simplified 26.0%

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

9. Final simplification 26.0%

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

Reproduce

herbie shell --seed 2024031 
(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)))))