Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 79.2%

Time: 26.2s

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: 66.6% 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: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-h}}{t_0}} \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -7.5e+110)
     (*
      (/ 1.0 (/ (sqrt (- h)) t_0))
      (*
       (/ 1.0 (sqrt (/ l d)))
       (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
     (if (<= h -2e-310)
       (*
        (sqrt (/ d h))
        (*
         (/ t_0 (sqrt (- l)))
         (- 1.0 (* 0.5 (pow (* (* 0.5 (/ D (/ d M))) (sqrt (/ h l))) 2.0)))))
       (*
        (fma (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5) 1.0)
        (/ d (* (sqrt l) (sqrt h))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -7.5e+110) {
		tmp = (1.0 / (sqrt(-h) / t_0)) * ((1.0 / sqrt((l / d))) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (h <= -2e-310) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 - (0.5 * pow(((0.5 * (D / (d / M))) * sqrt((h / l))), 2.0))));
	} else {
		tmp = fma(pow((D * (-0.5 * (M / d))), 2.0), ((h / l) * -0.5), 1.0) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -7.5e+110)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(-h)) / t_0)) * Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	elseif (h <= -2e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(0.5 * Float64(D / Float64(d / M))) * sqrt(Float64(h / l))) ^ 2.0)))));
	else
		tmp = Float64(fma((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -7.5e+110], N[(N[(1.0 / N[(N[Sqrt[(-h)], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.5e110

   1. Initial program 54.5%

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

   3. Simplified 60.0%

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

      1. clear-num 60.0%

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

      2. sqrt-div 59.9%

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

      3. metadata-eval 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. clear-num 60.0%

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

      2. sqrt-div 60.2%

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

      3. metadata-eval 60.2%

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

   7. Applied egg-rr 60.2%

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

      1. frac-2neg 60.2%

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

      2. sqrt-div 77.5%

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

   9. Applied egg-rr 77.5%

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

   if -7.5e110 < h < -1.999999999999994e-310

   1. Initial program 65.3%

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

   3. Simplified 62.8%

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

      1. add-sqr-sqrt 62.7%

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

      2. pow2 62.7%

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

      3. sqrt-prod 62.7%

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

      4. unpow2 62.7%

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

      5. sqrt-prod 26.7%

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

      6. add-sqr-sqrt 64.2%

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

      7. div-inv 64.2%

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

      8. metadata-eval 64.2%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in M around 0 66.8%

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

      1. associate-/l* 65.3%

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

   8. Simplified 65.3%

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

      1. frac-2neg 65.3%

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

      2. sqrt-div 75.6%

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

   10. Applied egg-rr 75.6%

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

   if -1.999999999999994e-310 < h

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

      1. pow1 66.7%

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

   5. Applied egg-rr 82.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}} \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2: 77.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

      1. clear-num 61.9%

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

      2. sqrt-div 61.9%

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

      3. metadata-eval 61.9%

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

   5. Applied egg-rr 61.9%

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

      1. clear-num 61.4%

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

      2. sqrt-div 62.2%

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

      3. metadata-eval 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. frac-2neg 62.2%

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

      2. sqrt-div 73.4%

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

   9. Applied egg-rr 73.4%

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

   if -1.999999999999994e-310 < h

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-udef 23.5%

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

   5. Applied egg-rr 28.8%

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

      1. expm1-def 45.5%

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

      2. expm1-log1p 82.0%

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

      3. associate-/r* 81.0%

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

      4. associate-*r* 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 77.7%

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

Alternative 3: 78.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

      1. clear-num 61.9%

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

      2. sqrt-div 61.9%

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

      3. metadata-eval 61.9%

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

   5. Applied egg-rr 61.9%

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

      1. clear-num 61.4%

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

      2. sqrt-div 62.2%

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

      3. metadata-eval 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. frac-2neg 62.2%

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

      2. sqrt-div 73.4%

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

   9. Applied egg-rr 73.4%

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

   if -1.999999999999994e-310 < h

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

      1. pow1 66.7%

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

   5. Applied egg-rr 82.0%

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

4. Final simplification 78.2%

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

Alternative 4: 77.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $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 d < -4.999999999999985e-310

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

      1. frac-2neg 61.9%

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

      2. sqrt-div 73.1%

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

   5. Applied egg-rr 73.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-udef 23.5%

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

   5. Applied egg-rr 28.8%

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

      1. expm1-def 45.5%

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

      2. expm1-log1p 82.0%

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

      3. associate-/r* 81.0%

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

      4. associate-*r* 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 77.5%

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

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.4 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 3.4e-282)
   (*
    (sqrt (/ d h))
    (*
     (- 1.0 (* 0.5 (pow (* (* 0.5 (/ D (/ d M))) (sqrt (/ h l))) 2.0)))
     (sqrt (/ d l))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ D d) (* 0.5 M)) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.4e-282) {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * pow(((0.5 * (D / (d / M))) * sqrt((h / l))), 2.0))) * sqrt((d / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 3.4d-282) then
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * (((0.5d0 * (d_1 / (d / m))) * sqrt((h / l))) ** 2.0d0))) * sqrt((d / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.4e-282) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * Math.pow(((0.5 * (D / (d / M))) * Math.sqrt((h / l))), 2.0))) * Math.sqrt((d / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 3.4e-282:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * math.pow(((0.5 * (D / (d / M))) * math.sqrt((h / l))), 2.0))) * math.sqrt((d / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * math.pow(((D / d) * (0.5 * M)), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 3.4e-282)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * (Float64(Float64(0.5 * Float64(D / Float64(d / M))) * sqrt(Float64(h / l))) ^ 2.0))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 3.4e-282)
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (((0.5 * (D / (d / M))) * sqrt((h / l))) ^ 2.0))) * sqrt((d / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * (((D / d) * (0.5 * M)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 3.4e-282], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[Power[N[(N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.39999999999999999e-282

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

      1. add-sqr-sqrt 62.1%

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

      2. pow2 62.1%

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

      3. sqrt-prod 62.1%

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

      4. unpow2 62.1%

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

      5. sqrt-prod 33.5%

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

      6. add-sqr-sqrt 64.0%

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

      7. div-inv 64.0%

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

      8. metadata-eval 64.0%

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

   5. Applied egg-rr 64.0%

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

   6. Taylor expanded in M around 0 64.8%

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

      1. associate-/l* 63.8%

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

   8. Simplified 63.8%

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

   if 3.39999999999999999e-282 < h

   1. Initial program 66.7%

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

      1. pow1 66.7%

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

   3. Applied egg-rr 81.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.4 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 6: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;h \leq 2.2 \cdot 10^{-286}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \left(D \cdot -0.5\right)\right)}^{2}, t_0, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)))
   (if (<= h 2.2e-286)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) t_0)))
     (*
      (fma (pow (* (/ M d) (* D -0.5)) 2.0) t_0 1.0)
      (/ (/ d (sqrt l)) (sqrt h))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double tmp;
	if (h <= 2.2e-286) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * t_0));
	} else {
		tmp = fma(pow(((M / d) * (D * -0.5)), 2.0), t_0, 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (h <= 2.2e-286)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * t_0)));
	else
		tmp = Float64(fma((Float64(Float64(M / d) * Float64(D * -0.5)) ^ 2.0), t_0, 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[h, 2.2e-286], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 2.1999999999999999e-286

   1. Initial program 61.8%

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

   3. Simplified 61.8%

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

      1. fma-udef 61.8%

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

      2. *-commutative 61.8%

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

      3. *-un-lft-identity 61.8%

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

      4. times-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. *-commutative 61.8%

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

   5. Applied egg-rr 61.8%

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

   if 2.1999999999999999e-286 < h

   1. Initial program 66.9%

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

   3. Simplified 67.0%

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

      1. expm1-log1p-u 34.3%

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

      2. expm1-udef 23.3%

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

   5. Applied egg-rr 28.7%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 82.8%

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

      3. associate-/r* 81.8%

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

      4. associate-*r* 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.2 \cdot 10^{-286}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \left(D \cdot -0.5\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.7 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.7e-136)
   (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
   (if (<= d -5e-310)
     (* d (log (exp (pow (* h l) -0.5))))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.7e-136) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else if (d <= -5e-310) {
		tmp = d * log(exp(pow((h * l), -0.5)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= (-6.7d-136)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else if (d <= (-5d-310)) then
        tmp = d * log(exp(((h * l) ** (-0.5d0))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.7e-136) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (d <= -5e-310) {
		tmp = d * Math.log(Math.exp(Math.pow((h * l), -0.5)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -6.7e-136:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	elif d <= -5e-310:
		tmp = d * math.log(math.exp(math.pow((h * l), -0.5)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.7e-136)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (d <= -5e-310)
		tmp = Float64(d * log(exp((Float64(h * l) ^ -0.5))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -6.7e-136)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	elseif (d <= -5e-310)
		tmp = d * log(exp(((h * l) ^ -0.5)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.7e-136], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Log[N[Exp[N[Power[N[(h * l), $MachinePrecision], -0.5], $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[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.6999999999999998e-136

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in M around 0 47.3%

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

      1. frac-2neg 72.5%

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

      2. sqrt-div 80.0%

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

   6. Applied egg-rr 56.2%

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

   if -6.6999999999999998e-136 < d < -4.999999999999985e-310

   1. Initial program 46.1%

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

      1. add-sqr-sqrt 46.0%

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

      2. pow2 46.0%

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

   3. Applied egg-rr 44.2%

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

   4. Taylor expanded in d around inf 11.2%

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

      1. unpow1/2 11.2%

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

      2. metadata-eval 11.2%

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

      3. pow-sqr 11.2%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{0.25} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{0.25}\right)} \]

      4. pow-sqr 11.2%

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

      5. metadata-eval 11.2%

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

      6. unpow1/2 11.2%

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

      7. unpow-1 11.2%

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

      8. sqr-pow 11.2%

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

      9. rem-sqrt-square 11.2%

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

      10. metadata-eval 11.2%

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

      11. sqr-pow 11.2%

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

      12. fabs-sqr 11.2%

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

      13. sqr-pow 11.2%

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

   6. Simplified 11.2%

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

      1. add-log-exp 28.6%

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

   8. Applied egg-rr 28.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Applied egg-rr 28.5%

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

      1. expm1-def 44.1%

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

      2. expm1-log1p 80.6%

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

      3. associate-/r* 80.1%

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

      4. associate-*r* 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*l* 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.7 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.2 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 2.2e-286)
   (*
    (sqrt (/ d h))
    (*
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))
     (sqrt (/ d l))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 2.2e-286) {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * sqrt((d / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 2.2d-286) then
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * sqrt((d / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 2.2e-286) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * Math.sqrt((d / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 2.2e-286:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * math.sqrt((d / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 2.2e-286)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * 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(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 2.2e-286)
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * sqrt((d / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 2.2e-286], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / 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[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.1999999999999999e-286

   1. Initial program 61.8%

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

   3. Simplified 61.8%

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

   if 2.1999999999999999e-286 < h

   1. Initial program 66.9%

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

   3. Applied egg-rr 28.4%

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

      1. expm1-def 43.6%

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

      2. expm1-log1p 81.4%

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

      3. associate-/r* 80.8%

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

      4. associate-*r* 80.8%

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

      5. *-commutative 80.8%

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

      6. associate-*l* 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.2 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 1.25 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1.25e-281)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (/ h l) (pow (/ D (* d (/ 2.0 M))) 2.0))))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.25e-281) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow((D / (d * (2.0 / M))), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 1.25d-281) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * ((d_1 / (d * (2.0d0 / m))) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.25e-281) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow((D / (d * (2.0 / M))), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 1.25e-281:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow((D / (d * (2.0 / M))), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1.25e-281)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D / Float64(d * Float64(2.0 / M))) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1.25e-281)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * ((D / (d * (2.0 / M))) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1.25e-281], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(d * N[(2.0 / M), $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[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.2499999999999999e-281

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

      1. clear-num 62.1%

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

      2. frac-times 61.3%

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

      3. *-un-lft-identity 61.3%

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

   5. Applied egg-rr 61.3%

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

   if 1.2499999999999999e-281 < h

   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. Applied egg-rr 28.6%

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

      1. expm1-def 43.9%

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

      2. expm1-log1p 81.3%

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

      3. associate-/r* 80.7%

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

      4. associate-*r* 80.7%

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

      5. *-commutative 80.7%

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

      6. associate-*l* 80.7%

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

   5. Simplified 80.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.25 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 10: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.3 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 3.3e-282)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.3e-282) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 3.3d-282) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 + (((d_1 * ((-0.5d0) * (m / d))) ** 2.0d0) * ((h / l) * (-0.5d0))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.3e-282) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 3.3e-282:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 3.3e-282)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 3.3e-282)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 3.3e-282], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $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[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.3e-282

   1. Initial program 62.1%

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

   3. Simplified 61.3%

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

      1. fma-udef 61.3%

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

      2. *-commutative 61.3%

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

      3. *-un-lft-identity 61.3%

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

      4. times-frac 61.3%

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

      5. metadata-eval 61.3%

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

      6. *-commutative 61.3%

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

   5. Applied egg-rr 61.3%

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

   if 3.3e-282 < h

   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. Applied egg-rr 28.6%

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

      1. expm1-def 43.9%

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

      2. expm1-log1p 81.3%

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

      3. associate-/r* 80.7%

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

      4. associate-*r* 80.7%

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

      5. *-commutative 80.7%

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

      6. associate-*l* 80.7%

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

   5. Simplified 80.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.3 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 5e-282)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (+ 1.0 (* (pow (* D (* -0.5 (/ M d))) 2.0) (* (/ h l) -0.5))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ D d) (* 0.5 M)) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5e-282) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 5d-282) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 + (((d_1 * ((-0.5d0) * (m / d))) ** 2.0d0) * ((h / l) * (-0.5d0))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5e-282) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 + (Math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((D / d) * (0.5 * M)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 5e-282:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 + (math.pow((D * (-0.5 * (M / d))), 2.0) * ((h / l) * -0.5)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * math.pow(((D / d) * (0.5 * M)), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 5e-282)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 + Float64((Float64(D * Float64(-0.5 * Float64(M / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 5e-282)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (((D * (-0.5 * (M / d))) ^ 2.0) * ((h / l) * -0.5)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * (((D / d) * (0.5 * M)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 5e-282], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(-0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.0000000000000001e-282

   1. Initial program 62.1%

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

   3. Simplified 61.3%

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

      1. fma-udef 61.3%

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

      2. *-commutative 61.3%

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

      3. *-un-lft-identity 61.3%

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

      4. times-frac 61.3%

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

      5. metadata-eval 61.3%

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

      6. *-commutative 61.3%

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

   5. Applied egg-rr 61.3%

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

   if 5.0000000000000001e-282 < h

   1. Initial program 66.7%

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

      1. pow1 66.7%

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

   3. Applied egg-rr 81.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(D \cdot \left(-0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 12: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.4 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.4e-139)
   (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
   (if (<= d -5e-310)
     (* d (log (exp (pow (* h l) -0.5))))
     (* d (* (pow l -0.5) (sqrt (/ 1.0 h)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.4e-139) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else if (d <= -5e-310) {
		tmp = d * log(exp(pow((h * l), -0.5)));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= (-8.4d-139)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else if (d <= (-5d-310)) then
        tmp = d * log(exp(((h * l) ** (-0.5d0))))
    else
        tmp = d * ((l ** (-0.5d0)) * sqrt((1.0d0 / h)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.4e-139) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (d <= -5e-310) {
		tmp = d * Math.log(Math.exp(Math.pow((h * l), -0.5)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.4e-139:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	elif d <= -5e-310:
		tmp = d * math.log(math.exp(math.pow((h * l), -0.5)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.sqrt((1.0 / h)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.4e-139)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (d <= -5e-310)
		tmp = Float64(d * log(exp((Float64(h * l) ^ -0.5))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.4e-139)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	elseif (d <= -5e-310)
		tmp = d * log(exp(((h * l) ^ -0.5)));
	else
		tmp = d * ((l ^ -0.5) * sqrt((1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.4e-139], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Log[N[Exp[N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.40000000000000033e-139

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in M around 0 47.3%

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

      1. frac-2neg 72.5%

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

      2. sqrt-div 80.0%

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

   6. Applied egg-rr 56.2%

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

   if -8.40000000000000033e-139 < d < -4.999999999999985e-310

   1. Initial program 46.1%

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

      1. add-sqr-sqrt 46.0%

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

      2. pow2 46.0%

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

   3. Applied egg-rr 44.2%

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

   4. Taylor expanded in d around inf 11.2%

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

      1. unpow1/2 11.2%

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

      2. metadata-eval 11.2%

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

      3. pow-sqr 11.2%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{0.25} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{0.25}\right)} \]

      4. pow-sqr 11.2%

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

      5. metadata-eval 11.2%

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

      6. unpow1/2 11.2%

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

      7. unpow-1 11.2%

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

      8. sqr-pow 11.2%

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

      9. rem-sqrt-square 11.2%

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

      10. metadata-eval 11.2%

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

      11. sqr-pow 11.2%

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

      12. fabs-sqr 11.2%

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

      13. sqr-pow 11.2%

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

   6. Simplified 11.2%

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

      1. add-log-exp 28.6%

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

   8. Applied egg-rr 28.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in d around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 39.4%

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

   6. Simplified 39.4%

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

      1. pow1/2 39.4%

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

      2. div-inv 39.4%

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

      3. metadata-eval 39.4%

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

      4. unpow-prod-down 48.4%

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

      5. metadata-eval 48.4%

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

      6. pow1/2 48.4%

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

      7. inv-pow 48.4%

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

      8. sqrt-pow1 48.5%

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

      9. metadata-eval 48.5%

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

      10. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. unpow1/2 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.4 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 13: 47.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4.5e-133)
   (* (/ 1.0 (sqrt (/ l d))) (sqrt (/ d h)))
   (if (<= d -5e-310)
     (* d (log (exp (pow (* h l) -0.5))))
     (* d (* (pow l -0.5) (sqrt (/ 1.0 h)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4.5e-133) {
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	} else if (d <= -5e-310) {
		tmp = d * log(exp(pow((h * l), -0.5)));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= (-4.5d-133)) then
        tmp = (1.0d0 / sqrt((l / d))) * sqrt((d / h))
    else if (d <= (-5d-310)) then
        tmp = d * log(exp(((h * l) ** (-0.5d0))))
    else
        tmp = d * ((l ** (-0.5d0)) * sqrt((1.0d0 / h)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4.5e-133) {
		tmp = (1.0 / Math.sqrt((l / d))) * Math.sqrt((d / h));
	} else if (d <= -5e-310) {
		tmp = d * Math.log(Math.exp(Math.pow((h * l), -0.5)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -4.5e-133:
		tmp = (1.0 / math.sqrt((l / d))) * math.sqrt((d / h))
	elif d <= -5e-310:
		tmp = d * math.log(math.exp(math.pow((h * l), -0.5)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.sqrt((1.0 / h)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4.5e-133)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * sqrt(Float64(d / h)));
	elseif (d <= -5e-310)
		tmp = Float64(d * log(exp((Float64(h * l) ^ -0.5))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -4.5e-133)
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	elseif (d <= -5e-310)
		tmp = d * log(exp(((h * l) ^ -0.5)));
	else
		tmp = d * ((l ^ -0.5) * sqrt((1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4.5e-133], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Log[N[Exp[N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.50000000000000009e-133

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in M around 0 47.3%

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

      1. clear-num 71.3%

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

      2. sqrt-div 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 48.5%

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

   if -4.50000000000000009e-133 < d < -4.999999999999985e-310

   1. Initial program 46.1%

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

      1. add-sqr-sqrt 46.0%

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

      2. pow2 46.0%

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

   3. Applied egg-rr 44.2%

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

   4. Taylor expanded in d around inf 11.2%

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

      1. unpow1/2 11.2%

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

      2. metadata-eval 11.2%

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

      3. pow-sqr 11.2%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{0.25} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{0.25}\right)} \]

      4. pow-sqr 11.2%

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

      5. metadata-eval 11.2%

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

      6. unpow1/2 11.2%

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

      7. unpow-1 11.2%

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

      8. sqr-pow 11.2%

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

      9. rem-sqrt-square 11.2%

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

      10. metadata-eval 11.2%

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

      11. sqr-pow 11.2%

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

      12. fabs-sqr 11.2%

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

      13. sqr-pow 11.2%

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

   6. Simplified 11.2%

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

      1. add-log-exp 28.6%

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

   8. Applied egg-rr 28.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in d around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 39.4%

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

   6. Simplified 39.4%

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

      1. pow1/2 39.4%

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

      2. div-inv 39.4%

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

      3. metadata-eval 39.4%

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

      4. unpow-prod-down 48.4%

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

      5. metadata-eval 48.4%

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

      6. pow1/2 48.4%

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

      7. inv-pow 48.4%

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

      8. sqrt-pow1 48.5%

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

      9. metadata-eval 48.5%

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

      10. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. unpow1/2 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 14: 39.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.35e-135)
   (sqrt (/ (pow d 2.0) (* h l)))
   (if (<= d -5e-310)
     (* d (cbrt (pow (/ 1.0 (* h l)) 1.5)))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.35e-135) {
		tmp = sqrt((pow(d, 2.0) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = d * cbrt(pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.35e-135) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.35e-135)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(h * l)));
	elseif (d <= -5e-310)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(h * l)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.35e-135], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[Power[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.34999999999999988e-135

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in d around inf 9.0%

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

      1. *-commutative 9.0%

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

      2. associate-/r* 9.0%

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

   6. Simplified 9.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 27.8%

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

      3. *-commutative 27.8%

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

      4. *-commutative 27.8%

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

      5. swap-sqr 23.8%

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

      6. add-sqr-sqrt 23.8%

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

      7. associate-/l/ 23.0%

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

      8. pow2 23.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell} \cdot \color{blue}{{d}^{2}}} \]

   8. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot {d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 23.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot {d}^{2}}{h \cdot \ell}}} \]

      2. *-lft-identity 23.0%

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

   10. Simplified 23.0%

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

   if -2.34999999999999988e-135 < d < -4.999999999999985e-310

   1. Initial program 46.1%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in d around inf 11.2%

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

      1. *-commutative 11.2%

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

      2. associate-/r* 11.2%

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

   6. Simplified 11.2%

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

      1. add-cbrt-cube 16.1%

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

      2. pow1/3 16.1%

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

      3. add-sqr-sqrt 16.1%

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

      4. pow1 16.1%

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

      5. pow1/2 16.1%

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

      6. metadata-eval 16.1%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot {\left(\frac{\frac{1}{\ell}}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 16.1%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \]

      8. associate-/l/ 16.1%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1 + \frac{1}{2}\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 16.1%

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

      10. metadata-eval 16.1%

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

   8. Applied egg-rr 16.1%

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

      1. unpow1/3 16.1%

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

   10. Simplified 16.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in d around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 39.4%

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

   6. Simplified 39.4%

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

      1. expm1-log1p-u 38.7%

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

      2. expm1-udef 23.8%

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

      3. sqrt-div 27.4%

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

      4. inv-pow 27.4%

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

      5. sqrt-pow1 27.4%

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

      6. metadata-eval 27.4%

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

   8. Applied egg-rr 27.4%

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

      1. expm1-def 47.5%

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

      2. expm1-log1p 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 15: 39.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.8e-133)
   (sqrt (/ (pow d 2.0) (* h l)))
   (if (<= d -5e-310)
     (* d (cbrt (pow (/ 1.0 (* h l)) 1.5)))
     (* d (* (pow l -0.5) (sqrt (/ 1.0 h)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-133) {
		tmp = sqrt((pow(d, 2.0) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = d * cbrt(pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-133) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else if (d <= -5e-310) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.8e-133)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(h * l)));
	elseif (d <= -5e-310)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(h * l)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.8e-133], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[Power[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7999999999999999e-133

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in d around inf 9.0%

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

      1. *-commutative 9.0%

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

      2. associate-/r* 9.0%

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

   6. Simplified 9.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 27.8%

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

      3. *-commutative 27.8%

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

      4. *-commutative 27.8%

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

      5. swap-sqr 23.8%

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

      6. add-sqr-sqrt 23.8%

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

      7. associate-/l/ 23.0%

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

      8. pow2 23.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell} \cdot \color{blue}{{d}^{2}}} \]

   8. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot {d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 23.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot {d}^{2}}{h \cdot \ell}}} \]

      2. *-lft-identity 23.0%

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

   10. Simplified 23.0%

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

   if -2.7999999999999999e-133 < d < -4.999999999999985e-310

   1. Initial program 46.1%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in d around inf 11.2%

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

      1. *-commutative 11.2%

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

      2. associate-/r* 11.2%

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

   6. Simplified 11.2%

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

      1. add-cbrt-cube 16.1%

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

      2. pow1/3 16.1%

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

      3. add-sqr-sqrt 16.1%

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

      4. pow1 16.1%

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

      5. pow1/2 16.1%

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

      6. metadata-eval 16.1%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot {\left(\frac{\frac{1}{\ell}}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 16.1%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \]

      8. associate-/l/ 16.1%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1 + \frac{1}{2}\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 16.1%

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

      10. metadata-eval 16.1%

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

   8. Applied egg-rr 16.1%

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

      1. unpow1/3 16.1%

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

   10. Simplified 16.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 66.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in d around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 39.4%

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

   6. Simplified 39.4%

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

      1. pow1/2 39.4%

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

      2. div-inv 39.4%

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

      3. metadata-eval 39.4%

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

      4. unpow-prod-down 48.4%

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

      5. metadata-eval 48.4%

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

      6. pow1/2 48.4%

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

      7. inv-pow 48.4%

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

      8. sqrt-pow1 48.5%

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

      9. metadata-eval 48.5%

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

      10. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. unpow1/2 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 16: 45.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 3.6e-298)
   (* (/ 1.0 (sqrt (/ l d))) (sqrt (/ d h)))
   (* d (* (pow l -0.5) (sqrt (/ 1.0 h))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.6e-298) {
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 3.6d-298) then
        tmp = (1.0d0 / sqrt((l / d))) * sqrt((d / h))
    else
        tmp = d * ((l ** (-0.5d0)) * sqrt((1.0d0 / h)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.6e-298) {
		tmp = (1.0 / Math.sqrt((l / d))) * Math.sqrt((d / h));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 3.6e-298:
		tmp = (1.0 / math.sqrt((l / d))) * math.sqrt((d / h))
	else:
		tmp = d * (math.pow(l, -0.5) * math.sqrt((1.0 / h)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 3.6e-298)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 3.6e-298)
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	else
		tmp = d * ((l ^ -0.5) * sqrt((1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 3.6e-298], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.60000000000000002e-298

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in M around 0 36.5%

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

      1. clear-num 61.7%

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

      2. sqrt-div 62.5%

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

      3. metadata-eval 62.5%

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

   6. Applied egg-rr 36.9%

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

   if 3.60000000000000002e-298 < h

   1. Initial program 66.4%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in d around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/r* 39.6%

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

   6. Simplified 39.6%

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

      1. pow1/2 39.6%

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

      2. div-inv 39.6%

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

      3. metadata-eval 39.6%

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

      4. unpow-prod-down 48.1%

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

      5. metadata-eval 48.1%

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

      6. pow1/2 48.1%

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

      7. inv-pow 48.1%

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

      8. sqrt-pow1 48.1%

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

      9. metadata-eval 48.1%

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

      10. metadata-eval 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. unpow1/2 48.1%

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

   10. Simplified 48.1%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 17: 45.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 3.6e-298)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (* d (* (pow l -0.5) (sqrt (/ 1.0 h))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.6e-298) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (h <= 3.6d-298) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = d * ((l ** (-0.5d0)) * sqrt((1.0d0 / h)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 3.6e-298) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 3.6e-298:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = d * (math.pow(l, -0.5) * math.sqrt((1.0 / h)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 3.6e-298)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 3.6e-298)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = d * ((l ^ -0.5) * sqrt((1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 3.6e-298], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.60000000000000002e-298

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in M around 0 36.5%

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

   if 3.60000000000000002e-298 < h

   1. Initial program 66.4%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in d around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/r* 39.6%

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

   6. Simplified 39.6%

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

      1. pow1/2 39.6%

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

      2. div-inv 39.6%

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

      3. metadata-eval 39.6%

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

      4. unpow-prod-down 48.1%

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

      5. metadata-eval 48.1%

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

      6. pow1/2 48.1%

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

      7. inv-pow 48.1%

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

      8. sqrt-pow1 48.1%

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

      9. metadata-eval 48.1%

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

      10. metadata-eval 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. unpow1/2 48.1%

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

   10. Simplified 48.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 18: 35.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.45 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.45e-138)
   (sqrt (/ (pow d 2.0) (* h l)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.45e-138) {
		tmp = sqrt((pow(d, 2.0) / (h * l)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= (-2.45d-138)) then
        tmp = sqrt(((d ** 2.0d0) / (h * l)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.45e-138) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.45e-138:
		tmp = math.sqrt((math.pow(d, 2.0) / (h * l)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.45e-138)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(h * l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.45e-138)
		tmp = sqrt(((d ^ 2.0) / (h * l)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.45e-138], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.45000000000000008e-138

   1. Initial program 70.1%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in d around inf 9.0%

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

      1. *-commutative 9.0%

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

      2. associate-/r* 9.0%

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

   6. Simplified 9.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 27.8%

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

      3. *-commutative 27.8%

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

      4. *-commutative 27.8%

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

      5. swap-sqr 23.8%

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

      6. add-sqr-sqrt 23.8%

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

      7. associate-/l/ 23.0%

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

      8. pow2 23.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell} \cdot \color{blue}{{d}^{2}}} \]

   8. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot {d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 23.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot {d}^{2}}{h \cdot \ell}}} \]

      2. *-lft-identity 23.0%

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

   10. Simplified 23.0%

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

   if -2.45000000000000008e-138 < d

   1. Initial program 62.3%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in d around inf 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-/r* 33.5%

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

   6. Simplified 33.5%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.45 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 19: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.9 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.9e-297)
   (sqrt (/ (pow d 2.0) (* h l)))
   (* d (/ (pow l -0.5) (sqrt h)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.9e-297) {
		tmp = sqrt((pow(d, 2.0) / (h * l)));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= 2.9d-297) then
        tmp = sqrt(((d ** 2.0d0) / (h * l)))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.9e-297) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.9e-297:
		tmp = math.sqrt((math.pow(d, 2.0) / (h * l)))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.9e-297)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(h * l)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.9e-297)
		tmp = sqrt(((d ^ 2.0) / (h * l)));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.9e-297], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.89999999999999989e-297

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in d around inf 10.3%

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

      1. *-commutative 10.3%

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

      2. associate-/r* 10.3%

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

   6. Simplified 10.3%

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

      1. add-sqr-sqrt 3.3%

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

      2. sqrt-unprod 21.2%

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

      3. *-commutative 21.2%

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

      4. *-commutative 21.2%

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

      5. swap-sqr 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. associate-/l/ 17.3%

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

      8. pow2 17.3%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell} \cdot \color{blue}{{d}^{2}}} \]

   8. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot {d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 17.3%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot {d}^{2}}{h \cdot \ell}}} \]

      2. *-lft-identity 17.3%

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

   10. Simplified 17.3%

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

   if 2.89999999999999989e-297 < d

   1. Initial program 66.4%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in d around inf 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.8%

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

   6. Simplified 39.8%

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

      1. expm1-log1p-u 39.1%

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

      2. expm1-udef 23.8%

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

      3. sqrt-div 27.5%

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

      4. inv-pow 27.5%

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

      5. sqrt-pow1 27.5%

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

      6. metadata-eval 27.5%

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

   8. Applied egg-rr 27.5%

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

      1. expm1-def 48.1%

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

      2. expm1-log1p 49.1%

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

   10. Simplified 49.1%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.9 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 20: 26.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 * sqrt(((1.0d0 / l) / h))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.6%

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

3. Simplified 64.6%

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

4. Taylor expanded in d around inf 26.0%

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

   1. *-commutative 26.0%

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

   2. associate-/r* 26.4%

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

6. Simplified 26.4%

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

7. Final simplification 26.4%

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

Alternative 21: 26.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 * l) ** (-0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.6%

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

   1. add-sqr-sqrt 64.6%

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

   2. pow2 64.6%

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

3. Applied egg-rr 66.1%

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

4. Taylor expanded in d around inf 26.0%

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

   1. unpow1/2 26.0%

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

   2. metadata-eval 26.0%

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

   3. pow-sqr 26.0%

      \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{0.25} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{0.25}\right)} \]

   4. pow-sqr 26.0%

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

   5. metadata-eval 26.0%

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

   6. unpow1/2 26.0%

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

   7. unpow-1 26.0%

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

   8. sqr-pow 26.0%

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

   9. rem-sqrt-square 26.1%

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

   10. metadata-eval 26.1%

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

   11. sqr-pow 26.1%

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

   12. fabs-sqr 26.1%

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

   13. sqr-pow 26.1%

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

6. Simplified 26.1%

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

7. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2023298 
(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)))))